Determining endpoint in etching processes using principal components analysis of optical emission spectra with thresholding

ABSTRACT

A method is provided for determining an etch endpoint. The method includes collecting intensity data representative of optical emission spectral wavelengths during a plasma etch process. The method further includes analyzing at least a portion of the collected intensity data into at most first and second Principal Components with respective Loadings and corresponding Scores. The method also includes determining the etch endpoint using the respective Loadings and corresponding Scores of the second Principal Component as an indicator for the etch endpoint using thresholding applied to the respective Loadings of the second Principal Component.

SPECIFIC REFERENCE TO PROVISIONAL APPLICATION

The present application claims priority to provisional applicationSerial No. 60/152,897, filed Sep. 8, 1999, and to provisionalapplication Serial No. 60/163,868, filed Nov. 5, 1999, the entire textsand figures of which are incorporated herein by reference withoutdisclaimer.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to semiconductor fabricationtechnology, and, more particularly, to determining the endpoint ofetching processes used during semiconductor fabrication.

2. Description of the Related Art

Typically, during semiconductor proceasing, an etching process, such asa reactive ion etch (RIE) process, is employed for etching fine linepatterns in a silicon wafer. RIE involves positioning a masked wafer ina chamber that contains plasma. The plasma contains etchant gases thatare dissociated in a radio frequency (RF) field so that reactive ionscontained in the etchant gases are vertically accelerated toward thewafer surface. The accelerated reactive ions combine chemically withunmasked material on the wafer surface. As a result, volatile etchproducts are produced. During such etching, single or multiple layers ofmaterial or films may be removed. Such material includes, for example,silicon dioxide (SiO₂), polysilicon (poly), and silicon nitride (Si₃N₄).Endpoint determination or detection refers to control of an etch stepand is useful in etching processes in general, and in RIE processes inparticular.

As a layer of unmasked material is etched, the volatile etch productsare incorporated into the plasma. As the RIE process approaches theinterface or end of the layer being etched, the amount of volatile etchproduct found in the plasma decreases since the amount of unmaskedmaterial being etched is reduced due to the etching. The amount ofvolatile etch product in the plasma may be tracked to determine theendpoint of the RIE process. In other words, the depletion or reductionin the amount of volatile etch product in the plasma during the RIEprocess typically can be used as an indicator for the end of the etchingprocess.

It is also possible to track a reactive species such as one of theetchant or input gases used to etch a layer of material. As the layer isetched, the reactive species will be depleted and relatively lowconcentrations of the reactive species will be found in the plasma.However, as more and more of the layer is consumed, the reactive specieswill be found in the plasma in increasingly higher concentrations. Atime trace of the optical emissions from such a reactive species willshow an increase in intensity as the layer is etched away. Tracking theintensity of a wavelength for a particular species using opticalemission spectroscopy (OES) may also be used for endpoint determinationor control of an etch process such as an RIE process.

Conventionally, OES has been used to track the amount of either volatileetch products or reactive species as a function of film thickness. Thesetechniques examine emissions from either the volatile etch products orreactive species in the plasma. As the film interface is reached duringetching, the emission species related to the etch of the film willeither decrease, in the case of volatile etch products, or increase, inthe case of reactive species.

More specifically, during an RIE process, plasma discharge materials,such as etchant, neutral, and reactive ions in the plasma, arecontinuously excited by electrons and collisions, giving off emissionsranging from ultraviolet to infrared radiation. An optical emissionspectrometer diffracts this light into its component wavelengths. Sinceeach species emits light at a wavelength characteristic only of thatspecies, it is possible to associate a certain wavelength with aparticular species, and to use this information to detect an etchendpoint.

As an example, in etching SiO₂ with CHF₃, carbon combines with oxygenfrom the wafer to form carbon monoxide (CO) as an etch product. It isknown that CO emits light at a wavelength of 451 nm, and that thiswavelength can be monitored for detecting the endpoint for such an etch.When the oxide is completely etched there is no longer a source ofoxygen and the CO peak at 451 nm decreases, thus signaling an etchendpoint.

In the above example, it is known that light emitted from CO at awavelength of 451 nm would be used for etch endpoint determination ordetection. However, such specific wavelength information is generallyunavailable, and it has been a formidable task to determine or select anappropriate wavelength to use for accurate etch endpoint determinationor control. This difficulty exists because of the numerous possibilitiesfor emissions. In other words, any molecule may emit light at amultitude of different wavelengths due to the many transition statesavailable for de-excitation. Therefore, given the process, the gasesutilized, and the material being etched, it is typically not readilyknown which wavelength in the spectrum to monitor for etch endpointdetermination or control. In this regard, the OES spectrum for a typicalRIE etch, for example, may be composed of hundreds, or even thousands,of wavelengths in the visible and ultraviolet bands.

Additionally, there is a trend towards using high-density plasma sourcesto replace RIE. One example is in the use of a high-density,inductively-coupled plasma (ICP). Another example is in the use ofelectron cyclotron resonance (ECR), which differs from RIE in plasmaformation. Generally, ECR operates at a lower pressure than aconventional RIE system, and is, therefore, able to etch finer linetrenches anisotropically. Comparison studies of the emissions fromhigh-density ICP, ECR and RIE plasmas show emphasis on different speciesand different wavelengths for similar input gas compositions. Theexcitation mechanisms and interactions of the particles at higherdensities and/or lower pressures are believed to account for many ofthese differences. Consequently, the experience and knowledgeaccumulated from RIE emissions may not carry over to high-density ICPemissions and ECR emissions. In other words, it may not be possible tomonitor the same species or wavelengths for etch endpoint determinationor detection in high-density ICP or ECR as were monitored for RIE, evenif similar materials are being etched using similar input gascompositions.

Conventional techniques for determining an endpoint in an etchingprocess using OES spectra are described, for example, in U.S. Pat. No.5,288,367, to Angell et al., entitled “End-point Detection,” and in U.S.Pat. No. 5,658,423, to Angell et al., entitled “Monitoring andControlling Plasma Processes via Optical Emission Using PrincipalComponent Analysis.” These conventional techniques typically stillentail singling out one wavelength to be used for signaling an etchendpoint, however. A conventional technique for effecting processcontrol by statistical analysis of the optical spectrum of a productproduced in a chemical process is described, for example, in U.S. Pat.No. 5,862,060, to Murray, Jr., entitled “Maintenance of process controlby statistical analysis of product optical spectrum” (the '060 patent).The '060 patent describes measuring the optical spectrum of each memberof a calibration sample set of selected products, determining byPrincipal Component Analysis (PCA) (or Partial Least Squares, PLS) notmore than four Principal Components to be used in the calibration sampleset, determining the differences in Scores of the Principal Componentsbetween a standard “target” product and a test product, and using thedifferences to control at least one process variable so as to minimizethe differences.

However, one drawback associated with conventional techniques fordetermining an endpoint in an etching process using PCA applied to OESspectra is the uncertainty of how many Principal Components to use inthe PCA analysis. This general question in conventional PCA applicationsis described, for example, in A User's Guide to Principal Components, byJ. Edward Jackson (Wiley Series in Probability and MathematicalStatistics, New York, 1991), particularly at pages 41-58. Typically, themore Principal Components that are used, the better the PCA approximatesthe system being analyzed, but the longer it takes to perform the PCA.For example, if all the Principal Components are used, the PCA exactlyreproduces the system being analyzed, but the full-Principal ComponentPCA takes the longest time to perform. However, determining the optimalnumber of PCA Principal Components to retain is also costly in terms oftime and resources involved.

The '060 patent describes, for example, that a very small number ofPrincipal Components, usually no more than 4, suffice to defineaccurately that sample spectrum space for the purpose of process controland that in some cases only 2 or 3 Principal Components need to be used.However, that still leaves an undesirable amount of uncertainty inwhether to use 2, 3 or 4 Principal Components. Furthermore, thisuncertainty can lead these conventional techniques to be cumbersome andslow and difficult to implement “on the fly” during real-time etchingprocesses, for example.

Moreover, modem state-of-the art OES systems are capable of collectingthousands of frequencies or wavelengths of optical emission spectraemanating from the glow discharge of gases in a plasma etch chamber.These wavelengths may be associated with the specific chemical speciesgenerated from entering reactant gases and their products. Theseproducts may result from gas phase reactions as well as reactions on thewafer and chamber wall surfaces. As the surface composition of the wafershifts from a steady-state etch of exposed surfaces to the completeremoval of the etched material, the wavelengths and frequencies of theoptical emission spectra also shift. Detection of this shift may allowfor etch endpoint determination, indicating the completion of therequired etch. Detection of this shift also may allow for termination ofthe etch process before deleterious effects associated with an over-etchcan occur. However, the sheer number of OES frequencies or wavelengthsavailable to monitor to determine an etch endpoint makes the problem ofselecting the appropriate OES frequencies or wavelengths to monitor evenmore severe.

An appropriate etch endpoint signal should have a high signal-to-noiseratio and be reproducible over the variations of the incoming wafers andthe state of the processing chamber, for example, whether or not theinternal hardware in the processing chamber is worn or new, or whetheror not the processing chamber is in a relatively “clean” or relatively“dirty” condition. Further, in particular applications, for example, theetching of contact and/or via holes, the signal-to-noise ratio mayinherently be very low simply by virtue of the small percentage ofsurface area being etched, typically about 1% or so.

The present invention is directed to overcoming, or at least reducingthe effects of, one or more of the problems set forth above.

SUMMARY OF THE INVENTION

In one aspect of the present invention, a method is provided fordetermining an etch endpoint. The method includes collecting intensitydata representative of optical emission spectral wavelengths during aplasma etch process. The method further includes analyzing at least aportion of the collected intensity data into at most first and secondPrincipal Components with respective Loadings and corresponding Scores.The method also includes determining the etch endpoint using therespective Loadings and corresponding Scores of the second PrincipalComponent as an indicator for the etch endpoint using thresholdingapplied to the respective Loadings of the second Principal Component.

In another aspect of the present invention, a computer-readable, programstorage device is provided, encoded with instructions that, whenexecuted by a computer, perform a method, the method includingcollecting intensity data representative of optical emission spectralwavelengths during a plasma etch process. The method further includesanalyzing at least a portion of the collected intensity data into atmost first and second Principal Components with respective Loadings andcorresponding Scores. The method also includes determining the etchendpoint using the respective Loadings and corresponding Scores of thesecond Principal Component as an indicator for the etch endpoint usingthresholding applied to the respective Loadings of the second PrincipalComponent.

In yet another aspect of the present invention, a computer programmed toperform a method is provided, the method including collecting intensitydata representative of optical emission spectral wavelengths during aplasma etch process. The method further includes analyzing at least aportion of the collected intensity data into at most first and secondPrincipal Components with respective Loadings and corresponding Scores.The method also includes determining the etch endpoint using therespective Loadings and corresponding Scores of the second PrincipalComponent as an indicator for the etch endpoint using thresholdingapplied to the respective Loadings of the second Principal Component.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention may be understood by reference to the followingdescription taken in conjunction with the accompanying drawings, inwhich the leftmost significant digit(s) in the reference numeralsdenote(s) the first figure in which the respective reference numeralsappear, and in which:

FIGS. 1-7 schematically illustrate a flow diagram for variousembodiments of a method according to the present invention;

FIGS. 8-14 schematically illustrate a flow diagram for variousalternative embodiments of a method according to the present invention;

FIGS. 15-21 schematically illustrate a flow diagram for yet othervarious embodiments of a method according to the present invention;

FIGS. 22 and 23 schematically illustrate first and second PrincipalComponents for respective data sets;

FIG. 24 schematically illustrates OES spectrometer counts plottedagainst wavelengths;

FIG. 25 schematically illustrates a time trace of OES spectrometercounts at a particular wavelength;

FIG. 26 schematically illustrates representative mean-scaledspectrometer counts for OES traces of a contact hole etch plottedagainst wavelengths and time;

FIG. 27 schematically illustrates a time trace of Scores for the secondPrincipal Component used to determine an etch endpoint;

FIGS. 28 and 29 schematically illustrate geometrically PrincipalComponents Analysis for respective data sets;

FIG. 30 schematically illustrates a method for fabricating asemiconductor device practiced in accordance with the present invention;and

FIG. 31 schematically illustrates workpieces being processed using ahigh-density plasma (HDP) etch processing tool, using a plurality ofcontrol input signals, in accordance with the present invention.

While the invention is susceptible to various modifications andalternative forms, specific embodiments thereof have been shown by wayof example in the drawings and are herein described in detail. It shouldbe understood, however, that the description herein of specificembodiments is not intended to limit the invention to the particularforms disclosed, but on the contrary, the intention is to cover allmodifications, equivalents, and alternatives falling within the spiritand scope of the invention as defined by the appended claims.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

Illustrative embodiments of the invention are described below. In theinterest of clarity, not all features of an actual implementation aredescribed in this specification. It will of course be appreciated thatin the development of any such actual embodiment, numerousimplementation-specific decisions must be made to achieve thedevelopers' specific goals, such as compliance with system-related andbusiness-related constraints, which will vary from one implementation toanother. Moreover, it will be appreciated that such a development effortmight be complex and time-consuming, but would nevertheless be a routineundertaking for those of ordinary skill in the art having the benefit ofthis disclosure.

Using optical emission spectroscopy (OES) as an analytical tool inprocess control (such as in etch endpoint determination) affords theopportunity for on-line measurements in real time. A calibration set ofOES spectra bounding the acceptable process space within which aparticular property (such as the etch endpoint) is to be controlled maybe obtained by conventional means. Applying a multivariant statisticalmethod such as Principal Components Analysis (PCA) to the calibrationset of OES spectra affords a method of identifying the most importantcharacteristics (Principal Components and respective “Loadings”) of theset of OES spectra that govern the controlled property, and areinherently related to the process. Control then is effected by usingonly two of such characteristics (Principal Components and respectiveLoadings and corresponding Scores), which can be determined quickly andsimply from the measured spectra, as the control criteria to be appliedto the process as a whole. The result is a very effective way ofcontrolling a complex process using only 2 criteria (the first andsecond Principal Components and respective Loadings and correspondingScores) objectively determined from a calibration set, which can beapplied in real time and virtually continuously, resulting in awell-controlled process that is (ideally) invariant.

In particular, we have found that the second Principal Componentcontains a very robust, high signal-to-noise indicator for etch endpointdetermination. In various illustrative embodiments, PCA may be appliedto the OES data, either the whole spectrum or at least a portion of thewhole spectrum. If the engineer and/or controller knows that only aportion of the OES data contains useful information, PCA may be appliedonly to that portion, for example. The PCA derives two or more of thePrincipal Components and respective Loadings and corresponding Scores.The respective Loadings may be used to determine which wavelengths couldbe used to determine an etch endpoint and also what the appropriateweightings for such wavelengths could be. The corresponding Scores maybe used to detect the etch endpoint, for example, by following timetraces of the corresponding Scores.

In one illustrative embodiment of a method according to the presentinvention, as shown in FIGS. 1-7, described in more detail below,archived data sets of OES wavelengths (or frequencies), from wafers thathad previously been plasma etched, may be processed and the weightedlinear combination of the intensity data, representative of OESwavelengths (or frequencies) collected over time during the plasma etch,defined by the second Principal Component may be used to determine anetch endpoint. For example, PCA may be applied to the archived OES dataand the respective Loadings for the first two Principal Components maybe retained as model information. When new OES data are taken during anetch process, approximations to the first two Scores for the new OESdata are calculated by using the respective Loadings for the first twoPrincipal Components retained from the model information derived fromthe archived OES data. The approximations to the second Score(corresponding to the second Principal Component) may be used todetermine the etch endpoint, by plotting the approximation to the secondScore as a time trace and looking for an abrupt change in the value ofthe approximation to the second Score on the time trace, for example.

In another illustrative embodiment of a method according to the presentinvention, as shown in FIGS. 8-14, described in more detail below, onlythe OES wavelengths (or frequencies) above a certain threshold value inweighting, as defined by the second Principal Component, may be used todetermine an etch endpoint. For example, instead of using Loadings forall wavelengths sampled, the Loadings may be compared with a certainthreshold value (such as approximately 0.03) and only those wavelengthswith loadings greater than or equal to the threshold value may be usedto determine the etch endpoint. In this way, relatively unimportantwavelengths may be ignored and the robustness of this embodiment may befurther enhanced.

In yet another illustrative embodiment of a method according to thepresent invention, as shown in FIGS. 15-21, described in more detailbelow, the determination of the second Principal Component may beperformed real-time in an early part of a plasma etching process, andthis determination of the second Principal Component may be used todetermine an etch endpoint. When used in real-time, the respective modelLoadings can be further adjusted based on real-time calculations. Theseadjustments can be wafer-to-wafer (during serial etching of a batch ofwafers) and/or within the etching of each single wafer. Forwafer-to-wafer adjustments, the Loadings from previous wafers may beused as the model Loadings. Within the etching of individual wafers, themodel Loadings may be calculated from an early portion of the plasmaetching process where the etching endpoint almost certainly will not beoccurring.

Any of these three embodiments may be applied in real-time etchprocessing. Alternatively, either of the first two illustrativeembodiments may be used as an identification technique when using batchetch processing, with archived data being applied statistically, todetermine an etch endpoint for the batch.

Various embodiments of the present invention are applicable to anyplasma etching process affording a characteristic data set whose qualitymay be said to define the “success” of the plasma etching process andwhose identity may be monitored by suitable spectroscopic techniquessuch as OES. The nature of the plasma etching process itself is notcritical, nor is the specific identity of the workpieces (such assemiconducting silicon wafers) whose spectra are being obtained.However, an “ideal” or “target” characteristic data set should be ableto be defined, having a known and/or determinable OES spectrum, andvariations in the plasma etching process away from the targetcharacteristic data set should be able to be controlled (ie., reduced)using identifiable independent process variables, e.g., etch endpoints,plasma energy and/or temperature, plasma chamber pressure, etchantconcentrations, flow rates, and the like. The ultimate goal of processcontrol is to maintain a properly operating plasma etching process atstatus quo. When a plasma etching process has been deemed to havereached a proper operating condition, process control should be able tomaintain it by proper adjustment of plasma etching process parameterssuch as etch endpoints, temperature, pressure, flow rate, residence timein the plasma chamber, and the like. At proper operating conditions, theplasma etching process affords the “ideal” or “target” characteristics.

One feature of illustrative embodiments of our control process is thatthe spectrum of the stream, hereafter referred to as the test stream,may be obtained continuously (or in a near-continuous manner) on-line,and compared to the spectrum of the “target” stream, hereafter referredto as the standard stream. The difference in the two spectra may then beused to adjust one or more of the process variables so as to produce atest stream more nearly identical with the standard stream. However, asenvisioned in practice, the complex spectral data will be reduced and/orcompressed to no more than 2 numerical values that define thecoordinates of the spectrum in the Principal Component space of theprocess subject to control. Typically small, incremental adjustmentswill be made so that the test stream approaches the standard streamwhile minimizing oscillation, particularly oscillations that will tendto lose process control rather than exercise control. Adjustments willbe made according to one or more suitable algorithms based on processmodeling, process experience, and/or artificial intelligence feedback,as described more fully within.

In principle, more than one process variable may be subject to control,although it is apparent that as the number of process variables undercontrol increases so does the complexity of the control process.Similarly, more than one test stream and more than one standard streammay be sampled, either simultaneously or concurrently, again with anincrease in complexity. In its simplest form, where there is one teststream and one process variable under control, one may analogize theforegoing to the use of a thermocouple in a reaction chamber to generatea heating voltage based on the sensed temperature, and to use thedifference between the generated heating voltage and a set point heatingvoltage to send power to the reaction chamber in proportion to thedifference between the actual temperature and the desired temperaturewhich, presumably, has been predetermined to be the optimum temperature.Since the result of a given change in a process variable can bedetermined quickly, this new approach opens up the possibility ofcontrolling the process by an automated “trial and error” feedbacksystem, since unfavorable changes can be detected quickly. Illustrativeembodiments of the present invention may operate as null-detectors withfeedback from the set point of operation, where the feedback signalrepresents the deviation of the total composition of the stream from atarget composition.

In one illustrative embodiment, OES spectra are taken of characteristicdata sets of plasma etching processes of various grades and quality,spanning the maximum range of values typical of the particular plasmaetching processes. Such spectra then are representative of the entirerange of plasma etching processes and are often referred to ascalibration samples. Note that because the characteristic data sets arerepresentative of those formed in the plasma etching processes, thecharacteristic data sets constitute a subset of those that define theboundaries of representative processes. It will be recognized that thereis no subset that is unique, that many different subsets may be used todefine the boundaries, and that the specific samples selected are notcritical.

Subsequently, the spectra of the calibration samples are subjected tothe well-known statistical technique of Principal Component Analysis(PCA) to afford a small number of Principal Components that largelydetermine the spectrum of any sample. The Principal Components, whichrepresent the major contributions to the spectral changes, are obtainedfrom the calibration samples by PCA or Partial Least Squares (PLS).Thereafter, any new sample may be assigned various contributions ofthese Principal Components that would reproduce the spectrum of the newsample. The amount of each Principal Component required is called aScore, and time traces of these Scores, tracking how various of theScores are changing with time, are used to detect deviations from the“target” spectrum.

In mathematical terms, a set of m time samples of an OES for a workpiece(such as a semiconductor wafer having various process layers formedthereon) taken at n channels or wavelengths or frequencies may bearranged as a rectangular n×m matrix X. In other words, the rectangularn×m matrix X may be comprised of 1 to n rows (each row corresponding toa separate OES channel or wavelength or frequency taken or sampled) and1 to m columns (each column corresponding to a separate OES spectrumtime sample). The values of the rectangular n×m matrix X may be countsrepresenting the intensity of the OES spectrum, or ratios of spectralintensities (normalized to a reference intensity), or logarithms of suchratios, for example. The rectangular n×m matrix X may have rank r, wherer≦min{m,n} is the maximum number of independent variables in the matrixX. The use of PCA, for example, generates a set of Principal ComponentsP (whose “Loadings,” or components, represent the contributions of thevarious spectral components) as an eigenmatrix (a matrix whose columnsare eigenvectors) of the equation ((X−M)(X−M)^(T))P=Λ²P, where M is arectangular n×m matrix of the mean values of the columns of X, Λ² is ann×n diagonal matrix of the squares of the eigenvalues λ_(i), i=1,2, . .. ,r, of the mean-scaled matrix X−M, and a Scores matrix, T, withX−M=PT^(T) and (X−M)^(T)=(PT^(T))_(T)=(T^(T))^(T)P^(T)=TP^(T), so that((X−M)(X−M)^(T))P=((PT^(T))(TP^(T)))P and((PT^(T))(TP^(T)))P=(P(T^(T)T)P^(T))P=P(T^(T)T)=Λ²P. The rectangular n×mmatrix X, also denoted X_(n×m), may have elements x_(ij), where i=1,2, .. . ,n, and j=1,2, . . . ,m, and the rectangular m×n matrix X^(T), thetranspose of the rectangular n×m matrix X, also denoted (X^(T))_(m×n),may have elements x_(ji), where i=1,2, . . . ,n, and j=1,2, . . . ,m.The n×n matrix (X−M)(X−M)^(T)) is the covariance matrix S_(n×n), havingelements s_(ij), where i=1,2, . . . ,n, and j=1,2, . . . ,n, defined sothat:${s_{ij} = \frac{{m{\sum\limits_{k = 1}^{m}{x_{ik}x_{jk}}}} - {\sum\limits_{k = 1}^{m}{x_{ik}{\sum\limits_{k = 1}^{m}x_{jk}}}}}{m\left( {m - 1} \right)}},$

corresponding to the rectangular n×m matrix X_(n×m).

For the purposes of the process control envisioned in this application,we have found that only 2 Principal Components are needed to accommodatethe data for a large range of plasma etching processes from a variety ofplasma etching chambers. The spectrum of the standard sample is thenexpressed in terms of time traces of the Scores of the 2 PrincipalComponents used, the spectrum of the test sample is similarly expressed,and the differences in the time traces of the Scores are used to controlthe process variables. Thus, no direct correlations between the samplespectrum and plasma etch endpoints need be known. In fact, the nature ofthe sample itself need not be known, as long as there is a standard, theOES spectrum of the standard is known, a set of at most 2 PrincipalComponents is identified for the class of test stream samples, and onecan establish how to use the 2 Principal Components to control theprocess variables (as discussed more fully below).

The spectrum of any sample may be expressed as a 2-dimensionalrepresentation of the intensity of emission at a particular wavelengthvs. the wavelength. That is, one axis represents intensity, the otherwavelength. The foregoing characterization of a spectrum is intended toincorporate various transformations that are mathematically covariant;e.g., instead of emission one might use absorption and/or transmission,and either may be expressed as a percentage or logarithmically. Whateverthe details, each spectrum may be viewed as a vector. The group ofspectra arising from a group of samples similarly corresponds to a groupof vectors. If the number of samples is N, there are at most N distinctspectra. If, in fact, none of the spectra can be expressed as a linearcombination of the other spectra, then the set of spectra define anN-dimensional spectrum space. However, in the cases of interest here,where a particular stream in an invariant plasma etching process isbeing sampled, we have observed that, in fact, any particular spectrummay be accurately represented as a linear combination of a small number,M=2, of other spectra—their “Principal Components” that we refer to as“working” spectra. These “working” spectra may be viewed as the newbasis set, i.e., linearly independent vectors that define the2-dimensional spectrum space in which the samples reside. The spectrumof any other sample is then a linear combination of the “working”spectra (or is at least projectable onto the 2-dimensional spectrumspace spanned by the “working” spectra). Our experience demonstratesthat the samples typically reside in, at most, a 2-dimensional spectrumspace, and that this 2-dimensional model suffices as a practical matter.

Statistical methods are available to determine the set of “working”spectra appropriate for any sample set, and the method of PCA is the onemost favored in the practice of the present invention, although othermethods, e.g., partial least squares, non-linear partial least squares,(or, with less confidence, multiple linear regression), also may beutilized. The “working” spectra, or the linearly independent vectorsdefining the sample spectrum space, are called Principal Components.Thus the spectrum of any sample is a linear combination of the PrincipalComponents. The fractional contribution of any Principal Component iscalled the Score for the respective Principal Component. Hence, thespectrum of any sample completely defines a set of Scores that greatlyreduces the apparent complexity of comparing different spectra. In fact,it has been found that for many processes of interest in semiconductorprocessing, a very small number of Principal Components, no more than 2,suffice to define accurately the sample spectrum space for the purposeof process control. This means that the process of characterizing thedifference between a test sample and the standard sample comes down tothe difference between only 2 numbers—the Scores of the respective 2Principal Components for the sample and “target.” It is significant tonote that the small number of Scores embodies a great deal ofinformation about the samples and the process, and that only 2 numbersare adequate to control the process within quite close tolerances. Byusing the null approach of illustrative embodiments of the presentinvention, the use of Scores is simplified to teaching small shifts (andrestoring them to zero) rather than drawing conclusions and/orcorrelations from the absolute values of the Scores.

Although other methods may exist, three methods for computing PrincipalComponents are as follows:

1. eigenanalysis (EIG);

2. singular value decomposition (SVD); and

3. nonlinear iterative partial least squares (NIPALS).

Each of the first two methods, EIG and SVD, simultaneously calculatesall possible Principal Components, whereas the NIPALS method allows forcalculation of one Principal Component at a time. However, an iterativeapproach to finding eigenvalues and eigenvectors, known as the powermethod, described more fully below, also allows for calculation of onePrincipal Component at a time. There are as many Principal Components asthere are channels (or wavelengths or frequencies). The power method mayefficiently use computing time.

For example, consider the 3×2 matrix A, its transpose, the 2×3 matrixA^(T), their 2×2 matrix product A^(T)A, and their 3×3 matrix productAA^(T): $A = \begin{pmatrix}1 & 1 \\1 & 0 \\1 & {- 1}\end{pmatrix}$ $A^{T} = \begin{pmatrix}1 & 1 & 1 \\1 & 0 & {- 1}\end{pmatrix}$ ${A^{T}A} = {{\begin{pmatrix}1 & 1 & 1 \\1 & 0 & {- 1}\end{pmatrix}\begin{pmatrix}1 & 1 \\1 & 0 \\1 & {- 1}\end{pmatrix}} = \begin{pmatrix}3 & 0 \\0 & 2\end{pmatrix}}$ ${AA}^{T} = {{\begin{pmatrix}1 & 1 \\1 & 0 \\1 & {- 1}\end{pmatrix}\begin{pmatrix}1 & 1 & 1 \\1 & 0 & {- 1}\end{pmatrix}} = {\begin{pmatrix}2 & 1 & 0 \\1 & 1 & 1 \\0 & 1 & 2\end{pmatrix}.}}$

EIG reveals that the eigenvalues λ of the matrix product A^(T)A are 3and 2. The eigenvectors of the matrix product A^(T)A are solutions t ofthe equation (A^(T)A)t=λt, and may be seen by inspection to be t₁^(T)=(1,0) and t₂ ^(T)=(0,1), belonging to the eigenvalues λ₁=3 andλ₂=2, respectively.

The power method, for example, may be used to determine the eigenvaluesλ and eigenvectors p of the matrix product AA^(T), where the eigenvaluesλ and the eigenvectors p are solutions p of the equation (AA^(T))p=λp. Atrial eigenvector p^(T)=(1,1,1) may be used:${\left( {AA}^{T} \right)\underset{\_}{p}} = {{\begin{pmatrix}2 & 1 & 0 \\1 & 1 & 1 \\0 & 1 & 2\end{pmatrix}\begin{pmatrix}1 \\1 \\1\end{pmatrix}} = {\begin{pmatrix}3 \\3 \\3\end{pmatrix} = {{3\begin{pmatrix}1 \\1 \\1\end{pmatrix}} = {\lambda_{1}{{\underset{\_}{p}}_{1}.}}}}}$

This indicates that the trial eigenvector p^(T)=(1,1,1) happened tocorrespond to the eigenvector p₁ ^(T)=(1,1,1) belonging to theeigenvalue λ₁=3. The power method then proceeds by subtracting the outerproduct matrix p₁p₁ ^(T) from the matrix product AA^(T) to form aresidual matrix R₁: $R_{1} = {{\begin{pmatrix}2 & 1 & 0 \\1 & 1 & 1 \\0 & 1 & 2\end{pmatrix} - {\begin{pmatrix}1 \\1 \\1\end{pmatrix}\begin{pmatrix}1 & 1 & 1\end{pmatrix}}} = {{\begin{pmatrix}2 & 1 & 0 \\1 & 1 & 1 \\0 & 1 & 2\end{pmatrix} - \begin{pmatrix}1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1\end{pmatrix}} = {\begin{pmatrix}1 & 0 & {- 1} \\0 & 0 & 0 \\{- 1} & 0 & 1\end{pmatrix}.}}}$

Another trial eigenvector p^(T)=(1,0,−1) may be used:${\left( {{AA}^{T} - {{\underset{\_}{p}}_{1}{\underset{\_}{p}}_{1}^{T}}} \right)\underset{\_}{p}} = {{R_{1}\underset{\_}{p}} = {{\begin{pmatrix}1 & 0 & {- 1} \\0 & 0 & 0 \\{- 1} & 0 & 1\end{pmatrix}\begin{pmatrix}1 \\0 \\{- 1}\end{pmatrix}} = {\begin{pmatrix}2 \\0 \\{- 2}\end{pmatrix} = {{2\begin{pmatrix}1 \\0 \\{- 1}\end{pmatrix}} = {\lambda_{2}{{\underset{\_}{p}}_{2}.}}}}}}$

This indicates that the trial eigenvector p^(T)=(1,0,−1) happened tocorrespond to the eigenvector p₂ ^(T)=(1,0,−1) belonging to theeigenvalue λ₂=2. The power method then proceeds by subtracting the outerproduct matrix p₂p₂ ^(T) from the residual matrix R₁ to form a secondresidual matrix R₂: $R_{2} = {{\begin{pmatrix}1 & 0 & {- 1} \\0 & 0 & 0 \\{- 1} & 0 & 1\end{pmatrix} - {\begin{pmatrix}1 \\0 \\{- 1}\end{pmatrix}\begin{pmatrix}1 & 0 & {- 1}\end{pmatrix}}} = {{\begin{pmatrix}1 & 0 & {- 1} \\0 & 0 & 0 \\{- 1} & 0 & 1\end{pmatrix} - \begin{pmatrix}1 & 0 & {- 1} \\0 & 0 & 0 \\{- 1} & 0 & 1\end{pmatrix}} = {\begin{pmatrix}0 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & 0\end{pmatrix}.}}}$

The fact that the second residual matrix R₂ vanishes indicates that theeigenvalue λ₃=0 and that the eigenvector p₃ is completely arbitrary. Theeigenvector p₃ may be conveniently chosen to be orthogonal to theeigenvectors p₁ ^(T)=(1,1,1) and p₂ ^(T)=(1,0,−1), so that theeigenvector p₃ ^(T)=(1,−2,1). Indeed, one may readily verify that:${\left( {AA}^{T} \right){\underset{\_}{p}}_{3}} = {{\begin{pmatrix}2 & 1 & 0 \\1 & 1 & 1 \\0 & 1 & 2\end{pmatrix}\begin{pmatrix}1 \\{- 2} \\1\end{pmatrix}} = {\begin{pmatrix}0 \\0 \\0\end{pmatrix} = {{0\begin{pmatrix}1 \\{- 2} \\1\end{pmatrix}} = {\lambda_{3}{{\underset{\_}{p}}_{3}.}}}}}$

Similarly, SVD of A shows that A=PT^(T), where P is the PrincipalComponent matrix and T is the Scores matrix: $A = {{\begin{pmatrix}{1/\sqrt{3}} & {1/\sqrt{2}} & {1/\sqrt{6}} \\{1/\sqrt{3}} & 0 & {{- 2}/\sqrt{6}} \\{1/\sqrt{3}} & {{- 1}/\sqrt{2}} & {1/\sqrt{6}}\end{pmatrix}\begin{pmatrix}\sqrt{3} & 0 \\0 & \sqrt{2} \\0 & 0\end{pmatrix}\begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}} = {\begin{pmatrix}{1/\sqrt{3}} & {1/\sqrt{2}} & {1/\sqrt{6}} \\{1/\sqrt{3}} & 0 & {{- 2}/\sqrt{6}} \\{1/\sqrt{3}} & {{- 1}/\sqrt{2}} & {1/\sqrt{6}}\end{pmatrix}{\begin{pmatrix}\sqrt{3} & 0 \\0 & \sqrt{2} \\0 & 0\end{pmatrix}.}}}$

SVD confirms that the singular values of A are 3 and 2, the positivesquare roots of the eigenvalues λ₁=3 and λ₂=2 of the matrix productA^(T)A. Note that the columns of the Principal Component matrix P arethe orthonormalized eigenvectors of the matrix product AA^(T).

Likewise, SVD of A^(T) shows that A^(T)=TP^(T):$A^{T} = {{\begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}\begin{pmatrix}\sqrt{3} & 0 & 0 \\0 & \sqrt{2} & 0\end{pmatrix}\begin{pmatrix}{1/\sqrt{3}} & {1/\sqrt{3}} & {1/\sqrt{3}} \\{1/\sqrt{2}} & 0 & {{- 1}/\sqrt{2}} \\{1/\sqrt{6}} & {{- 2}/\sqrt{6}} & {1/\sqrt{6}}\end{pmatrix}} = {A^{T} = {{\begin{pmatrix}\sqrt{3} & 0 & 0 \\0 & \sqrt{2} & 0\end{pmatrix}\begin{pmatrix}{1/\sqrt{3}} & {1/\sqrt{3}} & {1/\sqrt{3}} \\{1/\sqrt{2}} & 0 & {{- 1}/\sqrt{2}} \\{1/\sqrt{6}} & {{- 2}/\sqrt{6}} & {1/\sqrt{6}}\end{pmatrix}} = {{TP}^{T}.}}}}$

SVD confirms that the (non-zero) singular values of A^(T) are 3 and 2,the positive square roots of the eigenvalues λ₁=3 and λ₂=2 of the matrixproduct AA^(T). Note that the columns of the Principal Component matrixP (the rows of the Principal Component matrix P^(T)) are theorthonormalized eigenvectors of the matrix product AA^(T). Also notethat the non-zero elements of the Scores matrix T are the positivesquare roots 3 and 2 of the (non-zero) eigenvalues λ₁=3 and λ₂=2 of bothof the matrix products A^(T)A and AA^(T).

Taking another example, consider the 4×3 matrix B, its transpose, the3×4 matrix B^(T), their 3×3 matrix product B^(T)B, and their 4×4 matrixproduct BB^(T): $B = \begin{pmatrix}1 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\1 & {- 1} & 0\end{pmatrix}$ $B^{T} = \begin{pmatrix}1 & 1 & 1 & 1 \\1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0\end{pmatrix}$ ${B^{T}B} = {{\begin{pmatrix}1 & 1 & 1 & 1 \\1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0\end{pmatrix}\begin{pmatrix}1 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\1 & {- 1} & 0\end{pmatrix}} = \begin{pmatrix}4 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 2\end{pmatrix}}$ ${BB}^{T} = {{\begin{pmatrix}1 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\1 & {- 1} & 0\end{pmatrix}\begin{pmatrix}1 & 1 & 1 & 1 \\1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0\end{pmatrix}} = {\begin{pmatrix}2 & 1 & 1 & 0 \\1 & 2 & 0 & 1 \\1 & 0 & 2 & 1 \\0 & 1 & 1 & 2\end{pmatrix}.}}$

EIO reveals that the eigenvalues of the matrix product B^(T)B are 4, 2and 2. The eigenvectors of the matrix product B^(T)B are solutions t ofthe equation (B^(T)B)t=λt, and may be seen by inspection to be t₁^(T=()0,1,0), and t₃ ^(T)=(0,0,1), belonging to the eigenvalues λ₁=4,λ₂=2, and λ₃=2, respectively.

The power method, for example, may be used to determine the eigenvaluesλ and eigenvectors p of the matrix product BB^(T), where the eigenvaluesλ and the eigenvectors p are solutions p of the equation (BB^(T))p=λp. Atrial eigenvector p^(T)=(1,1,1,1) may be used:${\left( {BB}^{T} \right)\underset{\_}{p}} = {{\begin{pmatrix}2 & 1 & 1 & 0 \\1 & 2 & 0 & 1 \\1 & 0 & 2 & 1 \\0 & 1 & 1 & 2\end{pmatrix}\begin{pmatrix}1 \\1 \\1 \\1\end{pmatrix}} = {\begin{pmatrix}4 \\4 \\4 \\4\end{pmatrix} = {{4\begin{pmatrix}1 \\1 \\1 \\1\end{pmatrix}} = {\lambda_{1}{{\underset{\_}{p}}_{1}.}}}}}$

This indicates that the trial eigenvector p^(T)=(1,1,1,1) happened tocorrespond to the eigenvector p₁ ^(T)=(1,1,1,1) belonging to theeigenvalue λ₁=4. The power method then proceeds by subtracting the outerproduct matrix p₁p₁ ^(T) from the matrix product BB^(T) to form aresidual matrix R₁: $R_{1} = {{\begin{pmatrix}2 & 1 & 1 & 0 \\1 & 2 & 0 & 1 \\1 & 0 & 2 & 1 \\0 & 1 & 1 & 2\end{pmatrix} - \begin{pmatrix}1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 \\1 & 1 & 1 & 1\end{pmatrix}} = {\begin{pmatrix}1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0 \\0 & {- 1} & 1 & 0 \\{- 1} & 0 & 0 & 1\end{pmatrix}.}}$

Another trial eigenvector p^(T)=(1,0,0,−1) may be used:${\left( {{BB}^{T} - {{\underset{\_}{p}}_{1}{\underset{\_}{p}}_{1}^{T}}} \right)\underset{\_}{p}} = {{R_{1}\underset{\_}{p}} = {{\begin{pmatrix}1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0 \\0 & {- 1} & 1 & 0 \\{- 1} & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}1 \\0 \\0 \\{- 1}\end{pmatrix}} = {\begin{pmatrix}2 \\0 \\0 \\{- 2}\end{pmatrix} = {{2\begin{pmatrix}1 \\0 \\0 \\{- 1}\end{pmatrix}} = {\lambda_{2}{{\underset{\_}{p}}_{2}.}}}}}}$

This indicates that the trial eigenvector p^(T)=(1,0,0,−1) happened tocorrespond to the eigenvector p₂ ^(T)=(1,0,0,−1) belonging to theeigenvalue λ₂=2. The power method then proceeds by subtracting the outerproduct matrix p₂p₂ ^(T) from the residual matrix R₁ to form a secondresidual matrix R₂: $R_{2} = {{\begin{pmatrix}1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0 \\0 & {- 1} & 1 & 0 \\{- 1} & 0 & 0 & 1\end{pmatrix} - \begin{pmatrix}1 & 0 & 0 & {- 1} \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\{- 1} & 0 & 0 & 1\end{pmatrix}} = {\begin{pmatrix}0 & 0 & 0 & 0 \\0 & 1 & {- 1} & 0 \\0 & {- 1} & 1 & 0 \\0 & 0 & 0 & 0\end{pmatrix}.}}$

Another trial eigenvector p^(T)=(0,1,−1,0) may be used:${\left( {{BB}^{T} - {{\underset{\_}{p}}_{2}{\underset{\_}{p}}_{2}^{T}}} \right)\underset{\_}{p}} = {{R_{2}\underset{\_}{p}} = {{\begin{pmatrix}0 & 0 & 0 & 0 \\0 & 1 & {- 1} & 0 \\0 & {- 1} & 1 & 0 \\0 & 0 & 0 & 0\end{pmatrix}\begin{pmatrix}0 \\1 \\{- 1} \\0\end{pmatrix}} = {\begin{pmatrix}0 \\2 \\{- 2} \\0\end{pmatrix} = {{2\begin{pmatrix}0 \\1 \\{- 1} \\0\end{pmatrix}} = {\lambda_{3}{{\underset{\_}{p}}_{3}.}}}}}}$

This indicates that the trial eigenvector p^(T)=(0,1,−1,0) happened tocorrespond to the eigenvector p₃ ^(T)=(0,−1,0) belonging to theeigenvalue λ₃=2. The power method then proceeds by subtracting the outerproduct matrix p₃p₃ ^(T) from the second residual matrix R₂ to form athird residual matrix R₃: $R_{3} = {{\begin{pmatrix}0 & 0 & 0 & 0 \\0 & 1 & {- 1} & 0 \\0 & {- 1} & 1 & 0 \\0 & 0 & 0 & 0\end{pmatrix} - \begin{pmatrix}0 & 0 & 0 & 0 \\0 & 1 & {- 1} & 0 \\0 & {- 1} & 1 & 0 \\0 & 0 & 0 & 0\end{pmatrix}} = {\begin{pmatrix}0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{pmatrix}.}}$

The fact that the third residual matrix R₃ vanishes indicates that theeigenvalue λ₄ ₌₀ and that the eigenvector p₄ is completely arbitrary.The eigenvector p4 may be conveniently chosen to be orthogonal to theeigenvectors p₁ ^(T)=(1,1,1,1), p₂ ^(T)=(1,0,0,−1), and p₃^(T)=(0,1,−1), so that the eigenvector p₄ ^(T)=(1,−1,−1,1). Indeed, onemay readily verify that:${\left( {BB}^{T} \right){\underset{\_}{p}}_{4}} = {{\begin{pmatrix}2 & 1 & 1 & 0 \\1 & 2 & 0 & 1 \\1 & 0 & 2 & 1 \\0 & 1 & 1 & 2\end{pmatrix}\begin{pmatrix}1 \\{- 1} \\{- 1} \\1\end{pmatrix}} = {\begin{pmatrix}0 \\0 \\0 \\0\end{pmatrix} = {{0\begin{pmatrix}1 \\{- 1} \\{- 1} \\1\end{pmatrix}} = {\lambda_{4}{{\underset{\_}{p}}_{4}.}}}}}$

In this case, since the eigenvalues λ₂=2 and λ₃=2 are equal, and, hence,degenerate, the eigenvectors p₂ ^(T)=(1,0,0,−1) and p₃ ^(T)=(0,1,−1,0)belonging to the degenerate eigenvalues λ₂=2=λ₃ may be convenientlychosen to be orthonormal. A Gram-Schmidt orthonormalization proceduremay be used, for example.

Similarly, SVD of B shows that B=PT^(T), where P is the PrincipalComponent matrix and T is the Scores matrix: $B = {{\begin{pmatrix}{1/2} & {1/\sqrt{2}} & 0 & {1/2} \\{1/2} & 0 & {1/\sqrt{2}} & {{- 1}/2} \\{1/2} & 0 & {{- 1}/\sqrt{2}} & {{- 1}/2} \\{1/2} & {{- 1}/\sqrt{2}} & 0 & {1/2}\end{pmatrix}\begin{pmatrix}2 & 0 & 0 \\0 & \sqrt{2} & 0 \\0 & 0 & \sqrt{2} \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{pmatrix}} = {B = {{\begin{pmatrix}{1/2} & {1/\sqrt{2}} & 0 & {1/2} \\{1/2} & 0 & {1/\sqrt{2}} & {{- 1}/2} \\{1/2} & 0 & {{- 1}/\sqrt{2}} & {{- 1}/2} \\{1/2} & {{- 1}/\sqrt{2}} & 0 & {1/2}\end{pmatrix}\begin{pmatrix}2 & 0 & 0 \\0 & \sqrt{2} & 0 \\0 & 0 & \sqrt{2} \\0 & 0 & 0\end{pmatrix}} = {{PT}^{T}.}}}}$

SVD confirms that the singular values of B are 2, 2 and 2, the positivesquare roots of the eigenvalues λ₁=4, λ₂=2 and λ₃=2 of the matrixproduct B^(T)B.

Likewise, SVD of B^(T) shows that: $B^{T} = {{\begin{pmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{pmatrix}\begin{pmatrix}2 & 0 & 0 & 0 \\0 & \sqrt{2} & 0 & 0 \\0 & 0 & \sqrt{2} & 0\end{pmatrix}\begin{pmatrix}{1/2} & {1/2} & {1/2} & {1/2} \\{1/\sqrt{2}} & 0 & 0 & {{- 1}/\sqrt{2}} \\0 & {1/\sqrt{2}} & {{- 1}/\sqrt{2}} & 0 \\{1/2} & {{- 1}/2} & {{- 1}/2} & {1/2}\end{pmatrix}} = {B^{T} = {{\begin{pmatrix}2 & 0 & 0 & 0 \\0 & \sqrt{2} & 0 & 0 \\0 & 0 & \sqrt{2} & 0\end{pmatrix}\begin{pmatrix}{1/2} & {1/2} & {1/2} & {1/2} \\{1/\sqrt{2}} & 0 & 0 & {{- 1}/\sqrt{2}} \\0 & {1/\sqrt{2}} & {{- 1}/\sqrt{2}} & 0 \\{1/2} & {{- 1}/2} & {{- 1}/2} & {1/2}\end{pmatrix}} = {{TP}^{T}.}}}}$

SVD confirms that the (non-zero) singular values of B^(T) are 2, 2, and2, the positive square roots of the eigenvalues λ₁=4, λ₂=2 and λ₃=2 ofthe matrix product AA^(T). Note that the columns of the PrincipalComponent matrix P (the rows of the Principal Component matrix P^(T))are the orthonormalized eigenvectors of the matrix product BB^(T). Alsonote that the non-zero elements of the Scores matrix T are the positivesquare roots 2, 2, and 2 of the (non-zero) eigenvalues λ₁=4, λ₂=2 andλ₃=2 of both of the matrix products B^(T)B and BB^(T).

The matrices A and B discussed above have been used for the sake ofsimplifying the presentation of PCA and the power method, and are muchsmaller than the data matrices encountered in illustrative embodimentsof the present invention. For example, in one illustrative embodiment,for each wafer, 8 scans of OES data over 495 wavelengths may be takenduring an etching step, with about a 13 second interval between scans.In this illustrative embodiment, 18 wafers may be run and correspondingOES data collected. The data may be organized as a set of 18×495matrices X_(s)=(X_(ij))_(s), where s=1,2, . . . ,8, for each of thedifferent scans, and X_(ij) is the intensity of the ith wafer run at thejth wavelength. Putting all 8 of the 18×495 matrices X_(s)=X_(ij))_(s),for s=1,2, . . . ,8, next to each other produces the overall OES datamatrix X, an 18×3960 matrix X=[X₁,X₂, . . . ,X₈]=[(X_(ij))₁,(X_(ij))₂, .. . ,(X_(ij))₈]. Each row in X represents the OES data from 8 scans over495 wavelengths for a run. Brute force modeling using all 8 scans andall 495 wavelengths would entail using 3960 input variables to predictthe etching behavior of 18 sample wafers, an ill-conditioned regressionproblem. Techniques such as PCA and/or partial least squares (PLS, alsoknown as projection to latent structures) reduce the complexity in suchcases by revealing the hierarchical ordering of the data based on levelsof decreasing variability. In PCA, this involves finding successivePrincipal Components. In PLS techniques such as NIPALS, this involvesfinding successive latent vectors.

As shown in FIG. 22, a scatterplot 2200 of data points 2210 may beplotted in an n-dimensional variable space (n=3 in FIG. 22). The meanvector 2220 may lie at the center of a p-dimensional Principal Componentellipsoid 2230 (p=2 in FIG. 22). The mean vector 2220 may be determinedby taking the average of the columns of the overall OES data matrix X.The Principal Component ellipsoid 2230 may have a first PrincipalComponent 2240 (major axis in FIG. 22), with a length equal to thelargest eigenvalue of the mean-scaled OES data matrix X−M, and a secondPrincipal Component 2250 (minor axis in FIG. 22), with a length equal tothe next largest eigenvalue of the mean-scaled OES data matrix X−M.

For example, the 3×4 matrix B^(T) given above may be taken as theoverall OES data matrix X (again for the sake of simplicity),corresponding to 4 scans taken at 3 wavelengths. As shown in FIG. 23, ascatterplot 2300 of data points 2310 may be plotted in a 3-dimensionalvariable space. The mean vector 2320 μ may lie at the center of a2-dimensional Principal Component ellipsoid 2330 (really a circle, adegenerate ellipse). The mean vector 2320 μ may be determined by takingthe average of the columns of the overall OES 3×4 data matrix B^(T). ThePrincipal Component ellipsoid 2330 may have a first Principal Component2340 (“major” axis in FIG. 23) and a second Principal Component 2350(“minor” axis in FIG. 23). Here, the eigenvalues of the mean-scaled OESdata matrix B^(T)−M are equal and degenerate, so the lengths of the“major” and “minor” axes in FIG. 23 are equal. As shown in FIG. 23, themean vector 2320 μ is given by:${\underset{\_}{\mu} = {{\frac{1}{4}\left\lbrack {\begin{pmatrix}1 \\1 \\0\end{pmatrix} + \begin{pmatrix}1 \\0 \\1\end{pmatrix} + \begin{pmatrix}1 \\0 \\{- 1}\end{pmatrix} + \begin{pmatrix}1 \\{- 1} \\0\end{pmatrix}} \right\rbrack} = \begin{pmatrix}1 \\0 \\0\end{pmatrix}}},$

and the matrix M has the mean vector 2320 μ for all 4 columns.

In another illustrative embodiment, 5500 samples of each wafer may betaken on wavelengths between about 240-1100 nm at a high sample rate ofabout one per second. For example, 5551 sampling points/spectrum/second(corresponding to 1 scan per wafer per second taken at 5551 wavelengths,or 7 scans per wafer per second taken at 793 wavelengths, or 13 scansper wafer per second taken at 427 wavelengths, or 61 scans per wafer persecond taken at 91 wavelengths) may be collected in real time, duringetching of a contact hole using an Applied Materials AMAT 5300 Centuraetching chamber, to produce high resolution and broad band OES spectra.

As shown in FIG. 24, a representative OES spectrum 2400 of a contacthole etch is illustrated. Wavelengths, measured in nanometers (nm) areplotted along the horizontal axis against spectrometer counts plottedalong the vertical axis.

As shown in FIG. 25, a representative OES trace 2500 of a contact holeetch is illustrated. Time, measured in seconds (sec) is plotted alongthe horizontal axis against spectrometer counts plotted along thevertical axis. As shown in FIG. 25, by about 40 seconds into the etchingprocess, as indicated by dashed line 2510, the OES trace 2500 ofspectrometer counts “settles down” to a range of values less than orabout 300, for example.

As shown in FIG. 26, representative OES traces 2600 of a contact holeetch are illustrated. Wavelengths, measured in nanometers (nm) areplotted along a first axis, time, measured in seconds (sec) is plottedalong a second axis, and mean-scaled OES spectrometer counts, forexample, are plotted along a third (vertical) axis. As shown in FIG. 26,over the course of about 150 seconds of etching, three clusters ofwavelengths 2610, 2620 and 2630, respectively, show variations in therespective mean-scaled OES spectrometer counts. In one illustrativeembodiment, any one of the three clusters of wavelengths 2610, 2620 and2630 may be used, either taken alone or taken in any combination withany one (or both) of the others, as an indicator variable signaling anetch endpoint. In an alternative illustrative embodiment, only the twoclusters of wavelengths 2620 and 2630 having absolute values ofmean-scaled OES spectrometer counts that exceed a preselected thresholdabsolute mean-scaled OES spectrometer count value (for example, about200, as shown in FIG. 26) may be used, either taken alone or takentogether, as an indicator variable signaling an etch endpoint. In yetanother alternative illustrative embodiment, only one cluster ofwavelengths 2630 having an absolute value of mean-scaled OESspectrometer counts that exceeds a preselected threshold absolutemean-scaled OES spectrometer count value (for example, about 300, asshown in FIG. 26) may be used as an indicator variable signaling an etchendpoint.

As shown in FIG. 27, a representative Scores time trace 2700corresponding to the second Principal Component during a contact holeetch is illustrated. Time, measured in seconds (sec) is plotted alongthe horizontal axis against Scores (in arbitrary units) plotted alongthe vertical axis. As shown in FIG. 27, the Scores time trace 2700corresponding to the second Principal Component during a contact holeetch may start at a relatively high value initially, decrease with time,pass through a minimum value, and then begin increasing before levelingoff. We have found that the inflection point (indicated by dashed line2710, and approximately where the second derivative of the Scores timetrace 2700 with respect to time vanishes) is a robust indicator for theetch endpoint.

Principal Components Analysis (PCA) may be illustrated geometrically.For example, the 3×2 matrix C (similar to the 3×2 matrix A given above):$C = \begin{pmatrix}1 & {- 1} \\1 & 0 \\1 & 1\end{pmatrix}$

may be taken as the overall OES data matrix X (again for the sake ofsimplicity), corresponding to 2 scans taken at 3 wavelengths. As shownin FIG. 28, a scatterplot 2800 of OES data points 2810 and 2820, withcoordinates (1,1,1) and (−1,0,1), respectively, may be plotted in a3-dimensional variable space where the variables are respectivespectrometer counts for each of the 3 wavelengths. The mean vector 2830μ may lie at the center of a 1-dimensional Principal Component ellipsoid2840 (really a line, a very degenerate ellipsoid). The mean vector 2830μ may be determined by taking the average of the columns of the overallOES 3×2 matrix C. The Principal Component ellipsoid 2840 may have afirst Principal Component 2850 (the “major” axis in FIG. 28, with length5, lying along a first Principal Component axis 2860) and no second orthird Principal Component lying along second or third PrincipalComponent axes 2870 and 2880, respectively. Here, two of the eigenvaluesof the mean-scaled OES data matrix C−M are equal to zero, so the lengthsof the “minor” axes in FIG. 28 are both equal to zero. As shown in FIG.28, the mean vector 2830 μ is given by:${\underset{\_}{\mu} = {{\frac{1}{2}\left\lbrack {\begin{pmatrix}1 \\1 \\1\end{pmatrix} + \begin{pmatrix}{- 1} \\0 \\1\end{pmatrix}} \right\rbrack} = \begin{pmatrix}0 \\{1/2} \\1\end{pmatrix}}},$

and the matrix M has the mean vector 2830 μ for both columns. As shownin FIG. 28, PCA is nothing more than a principal axis rotation of theoriginal variable axes (here, the OES spectrometer counts for 3wavelengths) about the endpoint of the mean vector 2830 μ, withcoordinates (0,1/2,1) with respect to the original coordinate axes andcoordinates [0,0,0] with respect to the new Principal Component axes2860, 2870 and 2880. The Loadings are merely the direction cosines ofthe new Principal Component axes 2860, 2870 and 2880 with respect to theoriginal variable axes. The Scores are simply the coordinates of the OESdata points 2810 and 2820, [5^(0.5)/2,0,0] and [−5^(0.5)/2,0,0],respectively, referred to the new Principal Component axes 2860, 2870and 2880.

The mean-scaled 3×2 OES data matrix C−M, its transpose, the 2×3 matrix(C−M)^(T), their 2×2 matrix product (C−M)^(T)(C−M), and their 3×3 matrixproduct (C−M) (C−M)^(T) are given by: ${C - M} = {{\begin{pmatrix}1 & {- 1} \\1 & 0 \\1 & 1\end{pmatrix} - \begin{pmatrix}0 & 0 \\{1/2} & {1/2} \\1 & 1\end{pmatrix}} = \begin{pmatrix}1 & {- 1} \\{1/2} & {{- 1}/2} \\0 & 0\end{pmatrix}}$ $\left( {C - M} \right)^{T} = \begin{pmatrix}1 & {1/2} & 0 \\{- 1} & {{- 1}/2} & 0\end{pmatrix}$${\left( {C - M} \right)^{T}\left( {C - M} \right)} = {{\begin{pmatrix}1 & {1/2} & 0 \\{- 1} & {{- 1}/2} & 0\end{pmatrix}\begin{pmatrix}1 & {- 1} \\{1/2} & {{- 1}/2} \\0 & 0\end{pmatrix}} = \begin{pmatrix}{5/4} & {{- 5}/4} \\{{- 5}/4} & {5/4}\end{pmatrix}}$${\left( {C - M} \right)\left( {C - M} \right)^{T}} = {{\begin{pmatrix}1 & {- 1} \\{1/2} & {{- 1}/2} \\0 & 0\end{pmatrix}\begin{pmatrix}1 & {1/2} & 0 \\{- 1} & {{- 1}/2} & 0\end{pmatrix}} = \begin{pmatrix}2 & 1 & 0 \\1 & {1/2} & 0 \\0 & 0 & 0\end{pmatrix}}$

The 3×3 matrix (C−M)(C−M)^(T) is the covariance matrix S_(3×3), havingelements S_(ij), where i=1,2,3, and j=1,2,3, defined so that:${s_{ij} = \frac{{2{\sum\limits_{k = 1}^{2}{c_{ik}c_{jk}}}} - {\sum\limits_{k = 1}^{2}{c_{ik}{\sum\limits_{k = 1}^{2}c_{jk}}}}}{2\left( {2 - 1} \right)}},$

corresponding to the rectangular 3×2 matrix C_(3×2).

BIG reveals that the eigenvalues μ of the matrix product (C−M)^(T)(C−M)are 5/2 and 0, for example, by finding solutions to the secularequation: ${\begin{matrix}{{5/4} - \lambda} & {{- 5}/4} \\{{- 5}/4} & {{5/4} - \lambda}\end{matrix}} = 0.$

The eigenvectors of the matrix product (C−M)^(T)(C−M) are solutions t ofthe equation (C−M)^(T)(C−M)t=λt, which may be rewritten as((C−M)^(T)(C−M)−λ)t=0. For the eigenvalue λ₁=5/2, the eigenvector t₁ maybe seen by ${\begin{pmatrix}{{5/4} - \lambda} & {{- 5}/4} \\{{- 5}/4} & {{5/4} - \lambda}\end{pmatrix}\underset{\_}{t}} = {{\begin{pmatrix}{{- 5}/4} & {{- 5}/4} \\{{- 5}/4} & {{- 5}/4}\end{pmatrix}\underset{\_}{t}} = 0}$

to be t₁ ^(T)=(1,−1). For the eigenvalue λ₁=0, the eigenvector t₂ may beseen by ${\begin{pmatrix}{{5/4} - \lambda} & {{- 5}/4} \\{{- 5}/4} & {{5/4} - \lambda}\end{pmatrix}\underset{\_}{t}} = {{\begin{pmatrix}{5/4} & {{- 5}/4} \\{{- 5}/4} & {5/4}\end{pmatrix}\underset{\_}{t}} = {{0\quad {to}\quad {be}\quad {\underset{\_}{t}}_{2}^{T}} = {\left( {1,1} \right).}}}$

The power method, for example, may be used to determine the eigenvaluesλ and eigenvectors p of the matrix product (C−M)(C−M)^(T), where theeigenvalues λ and the eigenvectors p are solutions p of the equation((C−M)(C−M)^(T))p=λp. A trial eigenvector p^(T)=(1,1,1) may be used:${\left( {\left( {C - M} \right)\left( {C - M} \right)^{T}} \right)\underset{\_}{p}} = {{\begin{pmatrix}2 & 1 & 0 \\1 & {1/2} & 0 \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}1 \\1 \\1\end{pmatrix}} = {\begin{pmatrix}3 \\{3/2} \\0\end{pmatrix} = {{3\begin{pmatrix}\begin{matrix}1 \\{1/2}\end{matrix} \\0\end{pmatrix}} = {{3{\underset{\_}{q}\left( {\left( {C - M} \right)\left( {C - M} \right)^{T}} \right)}\underset{\_}{q}} = {{\begin{pmatrix}2 & 1 & 0 \\1 & {1/2} & 0 \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}1 \\{1/2} \\0\end{pmatrix}} = {\begin{pmatrix}{5/2} \\{5/4} \\0\end{pmatrix} = {{{5/2}\begin{pmatrix}\begin{matrix}1 \\{1/2}\end{matrix} \\0\end{pmatrix}} = {\lambda_{1}{{\underset{\_}{p}}_{1}.}}}}}}}}}$

This illustrates that the trial eigenvector p^(T)=(1,1,1) gets replacedby the improved trial eigenvector q^(T)=(1,1/2,0) that happened tocorrespond to the eigenvector p₁ ^(T)=(1,1/2,0) belonging to theeigenvalue λ₁=5/2. The power method then proceeds by subtracting theouter product matrix p₁p₁ ^(T) from the matrix product (C−M)(C−M)^(T) toform a residual matrix $R_{1} = {{\begin{pmatrix}2 & 1 & 0 \\1 & {1/2} & 0 \\0 & 0 & 0\end{pmatrix} - {\begin{pmatrix}1 \\{1/2} \\0\end{pmatrix}\begin{pmatrix}1 & {1/2} & 0\end{pmatrix}}} = {\begin{pmatrix}2 & 1 & 0 \\1 & {1/2} & 0 \\0 & 0 & 0\end{pmatrix} - \begin{pmatrix}1 & {1/2} & 0 \\{1/2} & {1/4} & 0 \\0 & 0 & 0\end{pmatrix}}}$ $R_{1} = {\begin{pmatrix}1 & {1/2} & 0 \\{1/2} & {1/4} & 0 \\0 & 0 & 0\end{pmatrix}.}$

Another trial eigenvector p^(T)=(−1,2,0), orthogonal to the eigenvectorp₁ ^(T)=(1,1/2,0) may be used:${\left( {{\left( {C - M} \right)\left( {C - M} \right)^{T}} - {{\underset{\_}{p}}_{1}{\underset{\_}{p}}_{1}^{T}}} \right)\underset{\_}{p}} = {{R_{1}\underset{\_}{p}} = {{\begin{pmatrix}1 & {1/2} & 0 \\{1/2} & {1/4} & 0 \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}{- 1} \\2 \\0\end{pmatrix}} = {\begin{pmatrix}0 \\0 \\0\end{pmatrix} = {{0\begin{pmatrix}\begin{matrix}{- 1} \\2\end{matrix} \\0\end{pmatrix}} = {\lambda_{2}{{\underset{\_}{p}}_{2}.}}}}}}$

This indicates that the trial eigenvector p^(T)=(−1,2,0) happened tocorrespond to the eigenvector p₂ ^(T)=(−1,2,0) belonging to theeigenvalue λ₂=0. The power method then proceeds by subtracting the outerproduct matrix p₂p₂ ^(T) from the residual matrix R₁ to form a secondresidual matrix R₂: $R_{2} = {{\begin{pmatrix}1 & {1/2} & 0 \\{1/2} & {1/4} & 0 \\0 & 0 & 0\end{pmatrix} - {\begin{pmatrix}{- 1} \\2 \\0\end{pmatrix}\begin{pmatrix}{- 1} & 2 & 0\end{pmatrix}}} = {\begin{pmatrix}1 & {1/2} & 0 \\{1/2} & {1/4} & 0 \\0 & 0 & 0\end{pmatrix} - \begin{pmatrix}1 & {- 2} & 0 \\{- 2} & 4 & 0 \\0 & 0 & 0\end{pmatrix}}}$ $R_{2} = {\begin{pmatrix}0 & {5/2} & 0 \\{5/2} & {{- 15}/4} & 0 \\0 & 0 & 0\end{pmatrix}.}$

Another trial eigenvector p^(T)=(0,0,1), orthogonal to the eigenvectorsp₁ ^(T)=(1,1/2,0) and p₂ ^(T)=(−1,2,0) may be used:${\left( {{\left( {C - M} \right)\left( {C - M} \right)^{T}} - {{\underset{\_}{p}}_{1}{\underset{\_}{p}}_{1}^{T}} - {{\underset{\_}{p}}_{2}{\underset{\_}{p}}_{2}^{T}}} \right)\underset{\_}{p}} = {{R_{2}\underset{\_}{p}} = {{\begin{pmatrix}1 & {5/2} & 0 \\{5/2} & {{- 15}/4} & 0 \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}0 \\0 \\1\end{pmatrix}} = \begin{pmatrix}0 \\0 \\0\end{pmatrix}}}$${\left( {{\left( {C - M} \right)\left( {C - M} \right)^{T}} - {{\underset{\_}{p}}_{1}{\underset{\_}{p}}_{1}^{T}} - {{\underset{\_}{p}}_{2}{\underset{\_}{p}}_{2}^{T}}} \right)\underset{\_}{p}} = {{R_{2}\underset{\_}{p}} = {{0\begin{pmatrix}0 \\0 \\1\end{pmatrix}} = {\lambda_{3}{{\underset{\_}{p}}_{3}.}}}}$

This indicates that the trial eigenvector p^(T)=(0,0,1) happened tocorrespond to the eigenvector p₃ ^(T)=(0,0,1) belonging to theeigenvalue λ₃=0. Indeed, one may readily verify that:${\left( {\left( {C - M} \right)\left( {C - M} \right)^{T}} \right){\underset{\_}{p}}_{3}} = {{\begin{pmatrix}2 & 1 & 0 \\1 & {1/2} & 0 \\0 & 0 & 0\end{pmatrix}\begin{pmatrix}0 \\0 \\1\end{pmatrix}} = {\begin{pmatrix}0 \\0 \\0\end{pmatrix} = {{0\begin{pmatrix}0 \\0 \\1\end{pmatrix}} = {\lambda_{3}{{\underset{\_}{p}}_{3}.}}}}}$

Similarly, SVD of C−M shows that C−M=PT^(T), where P is the PrincipalComponent matrix (whose columns are orthonormalized eigenvectorsproportional to p₁, p₂ and p₃, and whose elements are the Loadings, thedirection cosines of the new Principal Component axes 2860, 2870 and2880 related to the original variable axes) and T is the Scores matrix(whose rows are the coordinates of the OES data points 2810 and 2820,referred to the new Principal Component axes 2860, 2870 and 2880):${C - M} = {\begin{pmatrix}{2/\sqrt{5}} & {{- 1}/\sqrt{5}} & 0 \\{1/\sqrt{5}} & {2/\sqrt{5}} & 0 \\0 & 0 & 1\end{pmatrix}\begin{pmatrix}{\sqrt{5}/\sqrt{2}} & 0 \\0 & 0 \\0 & 0\end{pmatrix}\begin{pmatrix}{1/\sqrt{2}} & {{- 1}/\sqrt{2}} \\{1/\sqrt{2}} & {1/\sqrt{2}}\end{pmatrix}}$ ${C - M} = {{\begin{pmatrix}{2/\sqrt{5}} & {{- 1}/\sqrt{5}} & 0 \\{1/\sqrt{5}} & {2/\sqrt{5}} & 0 \\0 & 0 & 1\end{pmatrix}\begin{pmatrix}{\sqrt{5}/2} & {{- \sqrt{5}}/2} \\0 & 0 \\0 & 0\end{pmatrix}} = {{PT}^{T}.}}$

The transpose of the Scores matrix (T^(T)) is given by the product ofthe matrix of eigenvalues of C−M with a matrix whose rows areorthonormalized eigenvectors proportional to t₁ and t₂. As shown in FIG.28, the direction cosine (Loading) of the first Principal Component axis2860 with respect to the wavelength 1 counts axis is given by cosΘ₁₁=2/{square root over (5)}, and the direction cosine (Loading) of thefirst Principal Component axis 2860 with respect to the wavelength 2counts axis is given by cos Θ₂₁=1/{square root over (5)}. Similarly, thedirection cosine (Loading) of the first Principal Component axis 2860with respect to the wavelength 3 counts axis is given by cosΘ₃₁=cos(π/2)=0. Similarly, the direction cosine (Loading) of the secondPrincipal Component axis 2870 with respect to the wavelength 1 countsaxis is given by cos Θ₁₂=−1/{square root over (5)}, the direction cosine(Loading) of the second Principal Component axis 2870 with respect tothe wavelength 2 counts axis is given by cos Θ₂₂=2/{square root over(5)}, and the direction cosine (Loading) of the second PrincipalComponent axis 2870 with respect to the wavelength 3 counts axis isgiven by cos Θ₃₂=cos(π/2)=0. Lastly, the direction cosine (Loading) ofthe third Principal Component axis 2880 with respect to the wavelength 1counts axis is given by cos Θ₁₃=cos(π/2)=0, the direction cosine(Loading) of the third Principal Component axis 2880 with respect to thewavelength 2 counts axis is given by cos Θ₂₃=cos(π/2)=0, and thedirection cosine (Loading) of the third Principal Component axis 2880with respect to the wavelength 3 counts axis is given by cosΘ₃₃=cos(0)=1.

SVD confirms that the singular values of C−M are 5/2 and 0, thenon-negative square roots of the eigenvalues λ₁=5/2 and λ₂=0 of thematrix product (C−M)^(T)(C−M). Note that the columns of the PrincipalComponent matrix P are the orthonormalized eigenvectors of the matrixproduct (C−M)(C−M)^(T).

Taking another example, a 3×4 matrix D (identical to the 3×4 matrixB^(T) given above): $D = \begin{pmatrix}1 & 1 & 1 & 1 \\1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0\end{pmatrix}$

may be taken as the overall OES data matrix X (again for the sake ofsimplicity), corresponding to 4 scans taken at 3 wavelengths. As shownin FIG. 29, a scatterplot 2900 of OES data points with coordinates(1,1,0), (1,0,1), (1,0,−1) and (1,−1,0), respectively, may be plotted ina 3-dimensional variable space where the variables are respectivespectrometer counts for each of the 3 wavelengths. The mean vector 2920μmay lie at the center of a 2-dimensional Principal Component ellipsoid2930 (really a circle, a somewhat degenerate ellipsoid). The mean vector2920μ may be determined by taking the average of the columns of theoverall OES 3×4 matrix D. The Principal Component ellipsoid 2930 mayhave a first Principal Component 2940 (the “major” axis in FIG. 29, withlength 2, lying along a first Principal Component axis 2950), a secondPrincipal Component 2960 (the “minor” axis in FIG. 29, also with length2, lying along a second Principal Component axis 2970), and no thirdPrincipal Component lying along a third Principal Component axis 2980.Here, two of the eigenvalues of the mean-scaled OES data matrix D−M areequal, so the lengths of the “major” and “minor” axes of the PrincipalComponent ellipsoid 2930 in FIG. 29 are both equal, and the remainingeigenvalue is equal to zero, so the length of the other “minor” axis ofthe Principal Component ellipsoid 2930 in FIG. 29 is equal to zero. Asshown in FIG. 29, the mean vector 2920μ is given by:$\underset{\_}{\mu} = {{\frac{1}{4}\left\lbrack {\begin{pmatrix}1 \\1 \\0\end{pmatrix} + \begin{pmatrix}1 \\0 \\1\end{pmatrix} + \begin{pmatrix}1 \\0 \\{- 1}\end{pmatrix} + \begin{pmatrix}1 \\{- 1} \\0\end{pmatrix}} \right\rbrack} = \begin{pmatrix}1 \\0 \\0\end{pmatrix}}$

and the matrix M has the mean vector 2920μ for all 4 columns. As shownin FIG. 29, PCA is nothing more than a principal axis rotation of theoriginal variable axes (here, the OES spectrometer counts for 3wavelengths) about the endpoint of the mean vector 2920μ, withcoordinates (1,0,0) with respect to the original coordinate axes andcoordinates [0,0,0] with respect to the new Principal Component axes2950, 2970 and 2980. The Loadings are merely the direction cosines ofthe new Principal Component axes 2950, 2970 and 2980 with respect to theoriginal variable axes. The Scores are simply the coordinates of the OESdata points, [1,0,0], [0,1,0], [0,−1,0] and [−1,0,0], respectively,referred to the new Principal Component axes 2950, 2970 and 2980.

The 3×3 matrix product (D−M)(D−M)^(T) is given by:${\left( {D - M} \right)\left( {D - M} \right)^{T}} = {{\begin{pmatrix}0 & 0 & 0 & 0 \\1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0\end{pmatrix}\begin{pmatrix}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & {- 1} \\0 & {- 1} & 0\end{pmatrix}} = {\begin{pmatrix}0 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 2\end{pmatrix}.}}$

The 3×3 matrix (D−M)(D−M) is 3 times the covariance matrix S_(3×3),having elements s_(ij), where i=1,2,3, and j=1,2,3, defined so that:${s_{ij} = \frac{{4{\sum\limits_{k = 1}^{4}{d_{ik}d_{jk}}}} - {\sum\limits_{k = 1}^{4}{d_{ik}{\sum\limits_{k = 1}^{4}d_{jk}}}}}{4\left( {4 - 1} \right)}},$

corresponding to the rectangular 3×4 matrix D_(3×4).

EIG reveals that the eigenvalues of the matrix product (D−M)(D−M)7 are0, 2 and 2. The eigenvectors of the matrix product (D−M)(D−M)^(T) aresolutions p of the equation ((D−M)(D−M)^(T))p=λp, and may be seen byinspection to be p₁ ^(T)=(0,1,0), p₂ ^(T)=(0,0,1), and p₃ ^(T)=(1,0,0),belonging to the eigenvalues λ₁=2, λ₂=2, and λ₃=0, respectively(following the convention of placing the largest eigenvalue first).

As may be seen in FIG. 29, the direction cosine (Loading) of the firstPrincipal Component axis 2950 with respect to the wavelength 1 countsaxis is given by cos Θ₁₁=cos(π/2)=0, the direction cosine (Loading) ofthe first Principal Component axis 2970 with respect to the wavelength 2counts axis is given by cos Θ₂₁=cos(0)=1, and the direction cosine(Loading) of the first Principal Component axis 2860 with respect to thewavelength 3 counts axis is given by cos Θ₃₁=cos(π/2)=0. Similarly, thedirection cosine (Loading) of the second Principal Component axis 2970with respect to the wavelength 1 counts axis is given by cosΘ₁₂=cos(Θ/2)=0, the direction cosine (Loading) of the second PrincipalComponent axis 2970 with respect to the wavelength 2 counts axis isgiven by cos Θ₂₂=cos(π/2)=0, and the direction cosine (Loading) of thesecond Principal Component axis 2970 with respect to the wavelength 3counts axis is given by cos Θ₃₂=cos(0)=1. Lastly, the direction cosine(Loading) of the third Principal Component axis 2980 with respect to thewavelength 1 counts axis is given by cos Θ₁₃=cos(0)=1, the directioncosine (Loading) of the third Principal Component axis 2980 with respectto the wavelength 2 counts axis is given by cos Θ₂₃=cos(π/2)=0, and thedirection cosine (Loading) of the third Principal Component axis 2980with respect to the wavelength 3 counts axis is given by cosΘ₃₃=cos(π/2)=0.

The transpose of the Scores matrix T^(T) may be obtained simply bymultiplying the mean-scaled OES data matrix D−M on the left by thetranspose of the Principal Component matrix P, whose columns are p₁, p₂,p₃, the orthonormalized eigenvectors of the matrix product(D−M)(D−M)^(T):$T^{T} = {{P^{T}\left( {D - M} \right)} = {{\begin{pmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{pmatrix}\begin{pmatrix}0 & 0 & 0 & 0 \\1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0\end{pmatrix}} = {\begin{pmatrix}1 & 0 & 0 & {- 1} \\0 & 1 & {- 1} & 0 \\0 & 0 & 0 & 0\end{pmatrix}.}}}$

The columns of the transpose of the Scores matrix T^(T) (or,equivalently, the rows of the Scores matrix T) are, indeed, thecoordinates of the OES data points, [1,0,0], [0,1,0], [0,−1,0] and[−1,0,0], respectively, referred to the new Principal Component axes2950, 2970 and 2980.

We have found that the second Principal Component contains a veryrobust, high signal-to-noise indicator for etch endpoint determination.The overall mean-scaled OES rectangular n×m data matrix X_(nm)−M_(nm)may be decomposed into a portion corresponding to the first and secondPrincipal Components and respective Loadings and Scores, and a residualportion:${X - M} = {{{P_{PC}T_{PC}^{T}} + {P_{res}T_{res}^{T}}} = {{\begin{pmatrix}p_{1} & p_{2}\end{pmatrix}\begin{pmatrix}t_{1}^{T} \\t_{2}^{T}\end{pmatrix}} + {P_{res}T_{res}^{T}}}}$X − M = p₁t₁^(T) + p₂t₂^(T) + P_(res)T_(res)^(T) = X_(PC) + X_(res),

where P_(PC) is an n×2 matrix whose columns are the first and secondPrincipal Components, T_(PC) is an m×2 Scores matrix for the first andsecond Principal Components, T_(PC) ^(T) is a 2×m Scores matrixtranspose, P_(PC) ^(T) _(PC) ^(T)=X^(PC) is an n×m matrix, P_(res) is ann×(m−2) matrix whose columns are the residual Principal Components,T_(res) is an m×(m−2) Scores matrix for the residual PrincipalComponents, T_(res) ^(T) is an (m−2)×m Scores matrix transpose, andP_(res)T_(res) ^(T)=X_(res) is an n×m matrix. The kth column of X−M,x_(k), k=1,2, . . . ,m, an n×1 matrix, similarly decomposes intox_(k)=(x_(PC))_(k)+(x_(res))k, where (x_(PC))_(k)=P_(PC)P_(PC) ^(T)x_(k)is the projection of x_(k) into the Principal Component subspace (PCS)spanned by the first and second Principal Components,(x_(res))_(k)=(I_(n×n)−P_(PC)P_(PC) ^(T))x_(k) is the projection ofx_(k) into the residual subspace orthogonal to the PCS spanned by thefirst and second Principal Components, (x_(PC))_(k) ^(T)=x_(k)^(T)P_(PC)P_(PC) ^(T) is the projection of x_(k) ^(T) into the PrincipalComponent subspace (PCS) spanned by the first and second PrincipalComponents, (x_(res))_(k) ^(T)=x_(k) ^(T)(I_(n×n)−P_(PC)P_(PC) ^(T)) isthe projection of x_(k) ^(T) into the residual subspace orthogonal tothe PCS spanned by the first and second Principal Components, andI_(n×n) is the n×n identity matrix.

Using the projection of x_(k) into the PCS spanned by the first andsecond Principal Components and the projection of x_(k) into theresidual subspace orthogonal to the PCS spanned by the first and secondPrincipal Components, there are two tests of the amount of variance inthe data that are accounted for by the first and second PrincipalComponents. One test is the Q-test, also known as the Squared PredictionError (SPE) test, where SPE=∥(I_(n×n)−P_(PC)P_(PC)^(T))x_(k)∥²=∥(x_(res))_(k)∥²≦δ_(α) ². Another test is the Hotelling T²test, a multivariate generalization of the well-known Student's t-test,where T²=x_(k) ^(T)P_(PC)Λ⁻²P_(PC) ^(T)x_(k)=x_(k) ^(T)P_(PC)P_(PC)^(T)Λ⁻²P_(PC)P_(PC) ^(T)x_(k)=(x_(PC))_(k) ^(T)Λ⁻²(x_(PC))_(k)≦χ_(α) ²,where Λ² is a 2×2 diagonal matrix of the squares of the eigenvaluesλ_(i), i=1,2, belonging to the first and second Principal Components ofthe overall mean-scaled OES rectangular n×m data matrix X_(nm)−M_(nm).Both the SPE test and the Hotelling T² test can be used to monitor theetching process, for example.

We have also found that the first through fourth Principal Componentsare similarly useful for containing high signal-to-noise indicators foretch endpoint determination as well as being useful for OES datacompression. The overall mean-scaled OES rectangular n×m data matrixX_(nm)−M_(nm) may be decomposed into a portion corresponding to thefirst through fourth Principal Components and respective Loadings andScores, and a residual portion:${X - M} = {{{P_{PC}T_{PC}^{T}} + {P_{res}T_{res}^{T}}} = {{\begin{pmatrix}p_{1} & p_{2} & p_{3} & p_{4}\end{pmatrix}\begin{pmatrix}t_{1}^{T} \\t_{2}^{T} \\t_{3}^{T} \\t_{4}^{T}\end{pmatrix}} + {P_{res}T_{res}^{T}}}}$

which expands out to X−M=p₁t₁ ^(T)+p₂t₂ ^(T)+p₃t₃ ^(T)+p₄t₄^(T)+P_(res)T_(res) ^(T)=X_(PC)+X_(res), where P_(PC) is an n×4 matrixwhose columns are the first through fourth Principal Components, T_(PC)is an m×4 Scores matrix for the first through fourth PrincipalComponents, T_(PC) ^(T) is a 4×m Scores matrix transpose, P_(PC)T_(PC)^(T)=X_(PC) is an n×m matrix, P_(res) is an n×(m−4) matrix whose columnsare the residual Principal Components, T_(res) is an m×(m−4) Scoresmatrix for the residual Principal Components, T_(res) ^(T) is a (m−4)×mScores matrix transpose, and P_(res)T_(res) ^(T)=X_(res) is an n×mmatrix. The kth column of X−M, x_(k), k=1,2, . . . ,m, an n×1 matrix,similarly decomposes into x_(k)=(x_(PC))_(k)+(x_(res))_(k), where(x_(PC))_(k)=P_(PC)P_(PC) ^(T)x_(k) is the projection of x_(k) into thePrincipal Component subspace (PCS) spanned by the first through fourthPrincipal Components, (x_(res))_(k)=(I_(n×n)−P_(PC)P_(PC) ^(T))x_(k) isthe projection of x_(k) into the residual subspace orthogonal to the PCSspanned by the first through fourth Principal Components, (x_(PC))_(k)^(T)=x_(k) ^(T)P_(PC)P_(PC) ^(T) is the projection of x_(k) ^(T) intothe Principal Component subspace (PCS) spanned by the first throughfourth Principal Components, (x_(res))_(k) ^(T)=x_(k)^(T)(I_(n×n)−P_(PC)P_(PC) ^(T)) is the projection of x_(k) ^(T) into theresidual subspace orthogonal to the PCS spanned by the first throughfourth Principal Components, and I_(n×n) is the n×n identity matrix.

Using the projection of x_(k) into the PCS spanned by the first throughfourth Principal Components and the projection of x_(k) into theresidual subspace orthogonal to the PCS spanned by the first throughfourth Principal Components, there are again two tests of the amount ofvariance in the data that are accounted for by the first through fourthPrincipal Components. One test is the Q-test, also known as the SquaredPrediction Error (SPE) test, where SPE=∥(I_(n×n)−P_(PC)P_(PC)^(T))x_(k)∥²=∥(x_(res))_(k)∥²≦δ_(α) ². Another test is the Hotelling T²test, a multivariate generalization of the well-known Student's t-test,where T²=x_(k) ^(T)P_(PC)Λ⁻²P_(PC) ^(T)x_(k)=x_(k) ^(T)P_(PC)P_(PC)^(T)Λ⁻²P_(PC)P_(PC) ^(T)x_(k)=(x_(PC))_(k) ^(T)Λ⁻²(x_(PC)) _(k)≦χ_(α) ²,where Λ² is a 4×4 diagonal matrix of the squares of the eigenvaluesλ_(i), i=1,2,3,4 belonging to the first through fourth PrincipalComponents of the overall mean-scaled OES rectangular n×m data matrixX_(nm)−M_(nm).

More generally, the overall mean-scaled OES rectangular n×m data matrixX_(nm)−M_(nm) of rank r, where r≦min{m,n}, may be decomposed into aportion corresponding to the first through rth Principal Components andrespective Scores, and a residual portion:${X - M} = {{{P_{PC}T_{PC}^{T}} + {P_{res}T_{res}^{T}}} = {{\begin{pmatrix}p_{1} & p_{2} & \cdots & p_{r - 1} & p_{r}\end{pmatrix}\begin{pmatrix}t_{1}^{T} \\t_{2}^{T} \\\vdots \\t_{r - 1}^{T} \\t_{r}^{T}\end{pmatrix}} + {P_{res}T_{res}^{T}}}}$

which expands out to X−M=p₁t₁ ^(T)+p₂t₂ ^(T)+ . . . +p_(r−1)t_(r−1)^(T)+p_(r)t_(r) ^(T)+P_(res)T_(res) ^(T)=X_(PC)+X_(res), where P_(PC) isan n×r matrix whose columns are the first through rth PrincipalComponents, T_(PC) is an m×r Scores matrix for the first through rthPrincipal Components, T_(PC) ^(T) is an r×m Scores matrix transpose,P_(PC)T_(PC) ^(T)=X_(PC) is an n×m matrix, P_(res) is an n×(m−r) matrixwhose columns are the residual Principal Components (if m=r, P_(res)=0),T_(res) is an m×(m−r) Scores matrix for the residual PrincipalComponents (if m=r, T_(res)=0), T_(res) ^(T) is an (m−r)×m Scores matrixtranspose (if m=r, T_(res) ^(T)=0), and T_(res)P_(res) ^(T)=X_(res) isan n×m matrix (if m=r, X_(res)=0). The kth column of X−M, x_(k), k=1,2,. . . ,m, an n×1 matrix, similarly decomposes intox_(k)=(x_(PC))_(k)+(x_(res))_(k), where (x_(PC))_(k)=P_(PC)P_(PC)^(T)x_(k) is the projection of x^(k) into the Principal Componentsubspace (PCS) spanned by the first through rth Principal Components,(x_(res))_(k)=(I_(n×n)−P_(PC)P_(PC) ^(T))x_(k) is the projection ofx_(k) into the residual subspace orthogonal to the PCS spanned by thefirst through rth Principal Components, (x_(PC))_(k) ^(T)=x_(k)^(T)P_(PC)P_(PC) ^(T) is the projection of x_(k) ^(T) into the PrincipalComponent subspace (PCS) spanned by the first through rth PrincipalComponents, (x_(res))_(k) ^(T)=x_(k) ^(T)(I_(n×n)−P_(PC)P_(PC) ^(T)) isthe projection of x_(k) ^(T) into the residual subspace orthogonal tothe PCS spanned by the first through rth Principal Components, andI_(n×n) is the n×n identity matrix.

Using the projection of x_(k) into the PCS spanned by the first throughrth Principal Components and the projection of x_(k) into the residualsubspace orthogonal to the PCS spanned by the first through rthPrincipal Components, there are likewise two tests of the amount ofvariance in the data that are accounted for by the first through rthPrincipal Components. One test is the Q-test, also known as the SquaredPrediction Error (SPE) test, where SPE=∥(I_(n×n)−P_(PC)P_(PC)^(T))x_(k)∥²=∥(x_(res))_(k)∥²≦δ_(α) ². Another test is the Hotelling T²test, a multivariate generalization of the well-known Student's t-test,where T²=x_(k) ^(T)P_(PC)Λ⁻²P_(PC) ^(T)x_(k)=x_(k) ^(T)P_(PC)P_(PC)^(T)Λ⁻²P_(PC)P_(PC) ^(T)x_(k)=(x_(PC))_(k) ^(T)Λ⁻²(x_(PC))_(k)≦χ_(α) ²,where Λ² is an r×r diagonal matrix of the squares of the eigenvaluesλ_(i), i=1,2, . . . ,r belonging to the first through rth PrincipalComponents of the overall mean-scaled OES rectangular n×m data matrixX_(nm)−M_(nm) of rank r, where r≦min{m,n}.

In one illustrative embodiment of a method according to the presentinvention, as shown in FIGS. 1-7, archived data sets of OES wavelengths(or frequencies), from wafers that had previously been plasma etched,may be processed and the weighted linear combination of the intensitydata, representative of OES wavelengths (or frequencies) collected overtime during the plasma etch, defined by the second Principal Componentmay be used to determine an etch endpoint.

As shown in FIG. 1, a workpiece 100, such as a semiconducting substrateor wafer, having one or more process layers and/or semiconductor devicessuch as an MOS transistor disposed thereon, for example, is delivered toan etching preprocessing step j 105, where j may have any value from j=1to j=N−1. The total number N of processing steps, such as masking,etching, depositing material and the like, used to form the finishedworkpiece 100, may range from N=1 to about any finite value.

As shown in FIG. 2, the workpiece 100 is sent from the etchingpreprocessing step j 105 to an etching step j+1 110. In the etching stepj+1 110, the workpiece 100 is etched to remove selected portions fromone or more process layers formed in any of the previous processingsteps (such as etching preprocessing step j 105, where j may have anyvalue from j=1 to j=N−1). As shown in FIG. 2, if there is furtherprocessing to do on the workpiece 100 (if j<N−1), then the workpiece 100may be sent from the etching step j+1 110 and delivered to a postetchingprocessing step j+2 115 for further postetch processing, and then senton from the postetching processing step j+2 115. Alternatively, theetching step j+1 110 may be the final step in the processing of theworkpiece 100. In the etching step j+1 110, OES spectra are measured insitu by an OES spectrometer (not shown), producing OES data 120indicative of the state of the workpiece 100 during the etching.

In one illustrative embodiment, 5500 samples of each wafer may be takenon wavelengths between about 240-1100 nm at a high sample rate of aboutone per second. For example, 5551 sampling points/spectrum/second(corresponding to 1 scan per wafer per second taken at 5551 wavelengths)may be collected in real time, during etching of a contact hole using anApplied Materials AMAT 5300 Centura etching chamber, to produce highresolution and broad band OES spectra.

As shown in FIG. 3, the OES data 120 is sent from the etching step j+1110 and delivered to a first Principal Component (1st Factor)determination step 125. In the first Principal Component (1st Factor)determination step 125, the first Principal Component (1st Factor) and1st Scores, corresponding to the first Principal Component (1st Factor),may be determined from the OES data 120, for example, by any of thetechniques discussed above, producing the 1st Scores signal 130.

As shown in FIG. 4, the 1st Scores signal 130 is gent from the firstPrincipal Component (1st Loadings) determination step 125 to a secondPrincipal Component (2nd Loadings) determination step 135. In the secondPrincipal Component (2nd Loadings) determination step 135, the secondPrincipal Component (2nd Loadings) and 2nd Scores, corresponding to thesecond Principal Component (2nd Loadings), may be determined from theOES data 120 and the 1st Scores signal 130, for example, by any of thetechniques discussed above, producing the 2nd Scores signal 140.

As shown in FIG. 5, a feedback control signal 145 may be sent from thesecond Principal Component (2nd Loadings) determination step 135 to theetching step j+1 110 to adjust the processing performed in the etchingstep j+1 110. For example, based on the determination of the first andsecond Principal Components (1st and 2nd Loadings), and the 1st and 2ndScores corresponding to the first and second Principal Components (1stand 2nd Loadings), the feedback control signal 145 may be used to signalthe etch endpoint.

As shown in FIG. 6, in addition to, and/or instead of, the feedbackcontrol signal 135, the 2nd Scores signal 140 may be sent from thesecond Principal Component (2nd Loadings) determination step 135 and maythen be delivered to an archived PCA data comparison step 150. In thearchived PCA data comparison step 150, archived data sets of OESwavelengths (or frequencies), from wafers that had previously beenplasma etched, may be compared with the OES data 120 measured in situ inthe etching step 110, and the comparison may be used to determine anetch endpoint. In particular, first and second Principal Components (1stand 2nd Loadings), and the 1st and 2nd Scores corresponding to the firstand second Principal Components (1st and 2nd Loadings), previouslydetermined from the archived data sets of OES wavelengths (orfrequencies), may be compared with the first and second PrincipalComponents (1st and 2nd Loadings), and the 1st and 2nd Scorescorresponding to the first and second Principal Components (1st and 2ndLoadings), determined in the first Principal Component (1st Loadings)determination step 125 and the second Principal Component (2nd Loadings)determination step 135. This comparison of the respective PCAinformation may be used to determine an etch endpoint.

For example, PCA may be applied to the archived OES data and therespective Loadings for the first two Principal Components may beretained as model information. When new OES data are taken during anetch process, approximations to the first two Scores for the new OESdata are calculated by using the respective Loadings for the first twoPrincipal Components retained from the model information derived fromthe archived OES data. The approximations to the second Score(corresponding to the second Principal Component) may be used todetermine the etch endpoint, by plotting the approximation to the secondScore as a time trace and looking for an abrupt change in the value ofthe approximation to the second Score on the time trace, for example.

As shown in FIG. 7, a feedback control signal 155 may be sent from thearchived PCA data comparison step 150 to the etching step j+1 110 toadjust the processing performed in the etching step j+1 110. Forexample, based on the determination and comparison of the respectivefirst and second Principal Components (1st and 2nd Loadings), and the1st and 2nd Scores corresponding to the respective first and secondPrincipal Components (1st and 2nd Loadings), the feedback control signal155 may be used to signal the etch endpoint.

In another illustrative embodiment of a method according to the presentinvention, as shown in FIGS. 8-14, only the OES wavelengths (orfrequencies) above a certain threshold value in Loading or weighting, asdefined by the second Principal Component, may be used to determine anetch endpoint.

As shown in FIG. 8, a workpiece 800, such as a semiconducting substrateor wafer, having one or more process layers and/or semiconductor devicessuch as an MOS transistor disposed thereon, for example, is delivered toan etching preprocessing step j 805, where j may have any value from j=1to j=N−1. The total number N of processing steps, such as masking,etching, depositing material and the like, used to form the finishedworkpiece 800, may range from N=1 to about any finite value.

As shown in FIG. 9, the workpiece 800 is sent from the etchingpreprocessing step j 805 to an etching step j+1 810. In the etching stepj+1 810, the workpiece 800 is etched to remove selected portions fromone or more process layers formed in any of the previous processingsteps (such as etching preprocessing step j 805, where j may have anyvalue from j=1 to j=N−1). As shown in FIG. 9, if there is furtherprocessing to do on the workpiece 800 (if j<N−1), then the workpiece 800may be sent from the etching step j+1 810 and delivered to a postetchingprocessing step j+2 815 for further postetch processing, and then senton from the postetching processing step j+2 815. Alternatively, theetching step j+1 810 may be the final step in the processing of theworkpiece 800. In the etching step j+1 810, OES spectra are measured insitu by an OES spectrometer (not shown), producing OES data 820indicative of the state of the workpiece 800 during the etching.

As shown in FIG. 10, the OES data 820 is sent from the etching step j+1810 and delivered to a first Principal Component (1st Loadings)determination step 825. In the first Principal Component (1st Loadings)determination step 825, the first Principal Component (1st Loadings) and1st Scores, corresponding to the first Principal Component (1stLoadings), may be determined from the OES data 820, for example, by anyof the techniques discussed above, producing the 1st Scores signal 830.

As shown in FIG. 11, the 1st Scores signal 830 is sent from the firstPrincipal Component (1st Loadings) determination step 825 to a secondPrincipal Component (2nd Loadings) determination step 835. In the secondPrincipal Component (2nd Loadings) determination step 835, the secondPrincipal Component (2nd Loadings) and 2nd Scores, corresponding to thesecond Principal Component (2nd Loadings), may be determined from theOES data 820 and the 1st Scores signal 830, for example, by any of thetechniques discussed above, producing the 2nd Scores signal 840.

As shown in FIG. 12, a feedback control signal 845 may be sent from thesecond Principal Component (2nd Loadings) determination step 835 to theetching step j+1 810 to adjust the processing performed in the etchingstep j+1 810. For example, based on the determination of the first andsecond Principal Components (1st and 2nd Loadings), and the 1st and 2ndScores corresponding to the first and second Principal Components (1stand 2nd Loadings), the feedback control signal 845 may be used to signalthe etch endpoint.

As shown in FIG. 13, in addition to, and/or instead of, the feedbackcontrol signal 835, the 2nd Scores signal 840 may be sent from thesecond Principal Component (2nd Loadings) determination step 835 and maythen be delivered to a PCA data comparison with thresholding step 850.In the PCA data comparison with thresholding step 850, archived datasets of OES wavelengths (or frequencies), from wafers that hadpreviously been plasma etched, may be compared with the OES data 820,above a preselected threshold, measured in situ in the etching step 810,and the comparison with thresholding may be used to determine an etchendpoint. For example, instead of using Loadings for all wavelengthssampled, the Loadings may be compared with a certain threshold value(such as approximately 0.03) and only those wavelengths with loadingsgreater than or equal to the threshold value may be used to determinethe etch endpoint. In this way, relatively unimportant wavelengths maybe ignored and the robustness of this embodiment may be furtherenhanced.

As shown in FIG. 14, a feedback control signal 855 may be sent from thePCA data comparison with thresholding step 850 to the etching step j+1810 to adjust the processing performed in the etching step j+1 810. Forexample, based on the determination and comparison of the respectivefirst and second Principal Components (1st and 2nd Loadings), and the1st and 2nd Scores corresponding to the respective first and secondPrincipal Components (1st and 2nd Loadings), the feedback control signal855 may be used to signal the etch endpoint.

In yet another illustrative embodiment of a method according to thepresent invention, as shown in FIGS. 15-21, the determination of thesecond Principal Component may be performed real-time in an early partof a plasma etching process, and this determination of the secondPrincipal Component may be used to determine an etch endpoint.

As shown in FIG. 15, a workpiece 1500, such as a semiconductingsubstrate or wafer, having one or more process layers and/orsemiconductor devices such as an MOS transistor disposed thereon, forexample, is delivered to an etching preprocessing step j 1505, where jmay have any value from j=1 to j=N−1. The total number N of processingsteps, such as masking, etching, depositing material and the like, usedto form the finished workpiece 1500, may range from N=1 to about anyfinite value.

As shown in FIG. 16, the workpiece 1500 is sent from the etchingpreprocessing step j 1505 to an etching step j+1 1510. In the etchingstep j+1 1510, the workpiece 1500 is etched to remove selected portionsfrom one or more process layers formed in any of the previous processingsteps (such as etching preprocessing step j 1505, where j may have anyvalue from j=1 to j=N−1). As shown in FIG. 16, if there is furtherprocessing to do on the workpiece 1500 (if j<N−1), then the workpiece1500 may be sent from the etching step j+1 1510 and delivered to apostetching processing step j+2 1515 for further postetch processing,and then sent on from the postetching processing step j+2 1515.Alternatively, the etching step j+1 1510 may be the final step in theprocessing of the workpiece 1500. In the etching step j+1 1510, OESspectra are measured in situ by an OES spectrometer (not shown),producing OES data 1520 indicative of the state of the workpiece 1500during the etching.

As shown in FIG. 17, the OES data 1520 is sent from the etching step j+11510 and delivered to a first Principal Component (1st Loadings)determination step 1525. In the first Principal Component (1st Loadings)determination step 1525, the first Principal Component (1st Loadings)and 1st Scores, corresponding to the first Principal Component (1stLoadings), may be determined from the OES data 1520, for example, by anyof the techniques discussed above, producing the 1st Scores signal 1530.

As shown in FIG. 18, the 1st Scores signal 1530 is sent from the firstPrincipal Component (1st Loadings) determination step 1525 to a secondPrincipal Component (2nd Loadings) determination step 1535. In thesecond Principal Component (2nd Loadings) determination step 1535, thesecond Principal Component (2nd Loadings) and 2nd Scores, correspondingto the second Principal Component (2nd Loadings), may be determined fromthe OES data 1520 and the 1st Scores signal 1530, for example, by any ofthe techniques discussed above, producing the 2nd Scores signal 1540.

As shown in FIG. 19, a feedback control signal 1545 may be sent from thesecond Principal Component (2nd Loadings) determination step 1535 to theetching step j+1 1510 to adjust the processing performed in the etchingstep j+1 1510. For example, based on the determination of the first andsecond Principal Components (1st and 2nd Loadings), and the 1st and 2ndScores corresponding to the first and second Principal Components (1stand 2nd Loadings), the feedback control signal 1545 may be used tosignal the etch endpoint.

As shown in FIG. 20, in addition to, and/or instead of, the feedbackcontrol signal 1535, the 2nd Scores signal 1540 may be sent from thesecond Principal Component (2nd Loadings) determination step 1535 andmay then be delivered to a real-time PCA data comparison step 1550. Inthe real-time PCA data comparison step 1550, the determination of thesecond Principal Component may be performed real-time in an early partof a plasma etching process, and this determination of the secondPrincipal Component may be used to determine an etch endpoint. Inparticular, first and second Principal Components (with respective 1stand 2nd Loadings), and the 1st and 2nd Scores corresponding to the firstand second Principal Components (with respective 1st and 2nd Loadings),previously determined from the data sets of OES wavelengths (orfrequencies) taken in an earlier part of the same plasma etching processin etching step j+1 1510, may be compared with the first and secondPrincipal Components (with respective 1st and 2nd Loadings), and the 1stand 2nd Scores corresponding to the first and second PrincipalComponents (with respective 1st and 2nd Loadings), taken in a later partof the same plasma etching process in etching step j+1 1510, determinedin the first Principal Component (1st Loadings) determination step 1525and the second Principal Component (2nd Loadings) determination step1535. This real-time comparison of the respective PCA information may beused to determine an etch endpoint.

For example, when used in real-time, respective model Loadings can befurther adjusted based on real-time calculations. These adjustments canbe wafer-to-wafer (during serial etching of a batch of wafers) and/orwithin the etching of each single wafer. For wafer-to-wafer adjustments,the Loadings from previous wafers may be used as the model Loadings.Within the etching of individual wafers, the model Loadings may becalculated from an early portion of the plasma etching process where theetching endpoint almost certainly will not be occurring.

As shown in FIG. 21, a feedback control signal 1555 may be sent from thereal-time PCA data comparison step 1550 to the etching step j+1 1510 toadjust the processing performed in the etching step j+1 1510. Forexample, based on the determination and comparison of the respectivefirst and second Principal Components (1st and 2nd Loadings), and the1st and 2nd Scores corresponding to the respective first and secondPrincipal Components (1st and 2nd Loadings), the feedback control signal1555 may be used to signal the etch endpoint.

FIG. 30 illustrates one particular embodiment of a method 3000 practicedin accordance with the present invention. FIG. 31 illustrates oneparticular apparatus 3100 with which the method 3000 may be practiced.For the sake of clarity, and to further an understanding of theinvention, the method 3000 shall be disclosed in the context of theapparatus 3100. However, the invention is not so limited and admits widevariation, as is discussed further below.

Referring now to both FIGS. 30 and 31, a batch or lot of workpieces orwafers 3105 is being processed through an etch processing tool 3110. Theetch processing tool 3110 may be any etch processing tool known to theart, such as Applied Materials AMAT 5300 Centura etching chamber,provided it includes the requisite control capabilities. The etchprocessing tool 3110 includes an etch processing tool controller 3115for this purpose. The nature and function of the etch processing toolcontroller 3115 will be implementation specific. For instance, the etchprocessing tool controller 3115 may control etch control inputparameters such as etch recipe control input parameters and etchendpoint control parameters, and the like. Four workpieces 3105 areshown in FIG. 31, but the lot of workpieces or wafers, ie., the “waferlot,” may be any practicable number of wafers from one to any finitenumber.

The method 3000 begins, as set forth in box 3020, by measuringparameters such as OES spectral data characteristic of the etchprocessing performed on the workpiece 3105 in the etch processing tool3110. The nature, identity, and measurement of characteristic parameterswill be largely implementation specific and even tool specific. Forinstance, capabilities for monitoring process parameters vary, to somedegree, from tool to tool. Greater sensing capabilities may permit widerlatitude in the characteristic parameters that are identified andmeasured and the manner in which this is done. Conversely, lessersensing capabilities may restrict this latitude.

Turning to FIG. 31, in this particular embodiment, the etch processcharacteristic parameters are measured and/or monitored by tool sensors(not shown). The outputs of these tool sensors are transmitted to acomputer system 3130 over a line 3120. The computer system 3130 analyzesthese sensor outputs to identify the characteristic parameters.

Returning, to FIG. 30, once the characteristic parameter is identifiedand measured, the method 3000 proceeds by modeling the measured andidentified characteristic parameter using PCA, as set forth in box 3030.The computer system 3130 in FIG. 31 is, in this particular embodiment,programmed to model the characteristic parameter using PCA. The mannerin which this PCA modeling occurs will be implementation specific.

In the embodiment of FIG. 31, a database 3135 stores a plurality of PCAmodels and/or archived PCA data sets that might potentially be applied,depending upon which characteristic parameter is identified. Thisparticular embodiment, therefore, requires some a priori knowledge ofthe characteristic parameters that might be measured. The computersystem 3130 then extracts an appropriate model from the database 3135 ofpotential models to apply to the identified characteristic parameters.If the database 3135 does not include an appropriate model, then thecharacteristic parameter may be ignored, or the computer system 3130 mayattempt to develop one, if so programmed. The database 3135 may bestored on any kind of computer-readable, program storage medium, such asan optical disk 3140, a floppy disk 3145, or a hard disk drive (notshown) of the computer system 3130. The database 3135 may also be storedon a separate computer system (not shown) that interfaces with thecomputer system 3130.

Modeling of the identified characteristic parameter may be implementeddifferently in alternative embodiments. For instance, the computersystem 3130 may be programmed using some form of artificial intelligenceto analyze the sensor outputs and controller inputs to develop a PCAmodel on-the-fly in a real-time PCA implementation. This approach mightbe a useful adjunct to the embodiment illustrated in FIG. 31, anddiscussed above, where characteristic parameters are measured andidentified for which the database 3135 has no appropriate model.

The method 3000 of FIG. 30 then proceeds by applying the PCA model todetermine an etch endpoint, as set forth in box 3040. Depending on theimplementation, applying the PCA model may yield either a new value forthe etch endpoint control parameter or a correction and/or update to theexisting etch endpoint control parameter. The new etch endpoint controlparameter is then formulated from the value yielded by the PCA model andis transmitted to the etch processing tool controller 3115 over the line3120. The etch processing tool controller 3115 then controls subsequentetch process operations in accordance with the new etch control inputs.

Some alternative embodiments may employ a form of feedback to improvethe PCA modeling of characteristic parameters. The implementation ofthis feedback is dependent on several disparate facts, including thetool's sensing capabilities and economics. One technique for doing thiswould be to monitor at least one effect of the PCA model'simplementation and update the PCA model based on the effect(s)monitored. The update may also depend on the PCA model. For instance, alinear model may require a different update than would a non-linearmodel, all other factors being the same.

As is evident from the discussion above, some features of the presentinvention are implemented in software. For instance, the acts set forthin the boxes 3020-3040 in FIG. 30 are, in the illustrated embodiment,software-implemented, in whole or in part. Thus, some features of thepresent invention are implemented as instructions encoded on acomputer-readable, program storage medium. The program storage mediummay be of any type suitable to the particular implementation. However,the program storage medium will typically be magnetic, such as thefloppy disk 3145 or the computer 3130 hard disk drive (not shown), oroptical, such as the optical disk 3140. When these instructions areexecuted by a computer, they perform the disclosed functions. Thecomputer may be a desktop computer, such as the computer 3130. However,the computer might alternatively be a processor embedded in the etchprocessing tool 3110. The computer might also be a laptop, aworkstation, or a mainframe in various other embodiments. The scope ofthe invention is not limited by the type or nature of the programstorage medium or computer with which embodiments of the invention mightbe implemented.

Thus, some portions of the detailed descriptions herein are, or may be,presented in terms of algorithms, functions, techniques, and/orprocesses. These terms enable those skilled in the art most effectivelyto convey the substance of their work to others skilled in the art.These terms are here, and are generally, conceived to be aself-consistent sequence of steps leading to a desired result. The stepsare those requiring physical manipulations of physical quantities.Usually, though not necessarily, these quantities take the form ofelectromagnetic signals capable of being stored, transferred, combined,compared, and otherwise manipulated.

It has proven convenient at times, principally for reasons of commonusage, to refer to these signals as bits, values, elements, symbols,characters, terms, numbers, and the like. All of these and similar termsare to be associated with the appropriate physical quantities and aremerely convenient labels applied to these quantities and actions. Unlessspecifically stated otherwise, or as may be apparent from thediscussion, terms such as “processing,” “computing,” “calculating,”“determining,” “displaying,” and the like, used herein refer to theaction(s) and processes of a computer system, or similar electronicand/or mechanical computing device, that manipulates and transformsdata, represented as physical (electromagnetic) quantities within thecomputer system's registers and/or memories, into other data similarlyrepresented as physical quantities within the computer system's memoriesand/or registers and/or other such information storage, transmissionand/or display devices.

Any of these illustrative embodiments may be applied in real-time etchprocessing. Alternatively, either of the illustrative embodiments shownin FIGS. 1-7 and 8-14 may be used as an identification technique whenusing batch etch processing, with archived data being appliedstatistically, to determine an etch endpoint for the batch.

In various illustrative embodiments, a process engineer may be providedwith advanced process data monitoring capabilities, such as the abilityto provide historical parametric data in a user-friendly format, as wellas event logging, real-time graphical display of both current processingparameters and the processing parameters of the entire run, and remote,i.e., local site and worldwide, monitoring. These capabilities mayengender more optimal control of critical processing parameters, such asthroughput accuracy, stability and repeatability, processingtemperatures, mechanical tool parameters, and the like. This moreoptimal control of critical processing parameters reduces thisvariability. This reduction in variability manifests itself as fewerwithin-run disparities, fewer run-to-run disparities and fewertool-to-tool disparities. This reduction in the number of thesedisparities that can propagate means fewer deviations in product qualityand performance. In such an illustrative embodiment of a method ofmanufacturing according to the present invention, a monitoring anddiagnostics system may be provided that monitors this variability andoptimizes control of critical parameters.

An etch endpoint determination signal as in any of the embodimentsdisclosed above may have a high signal-to-noise ratio and may bereproducible over the variations of the incoming wafers and the state ofthe processing chamber, for example, whether or not the internalhardware in the processing chamber is worn or new, or whether or not theprocessing chamber is in a “clean” or a “dirty” condition. Further, inparticular applications, for example, the etching of contact and/or viaholes, an etch endpoint determination signal as in any of theembodiments disclosed above may have a high enough signal-to-noise ratioto overcome the inherently very low signal-to-noise ratio that may arisesimply by virtue of the small percentage (1% or so) of surface areabeing etched. In various illustrative embodiments, the etch endpointsignal becomes very stable, and may throughput may be improved byreducing the main etch time from approximately 145 seconds, for example,to approximately 90-100 seconds, depending on the thickness of theoxide. In the absence of an etch endpoint determination signal as in anyof the embodiments disclosed above, a longer etch time is conventionallyneeded to insure that all the material to be etched away has beenadequately etched away, even in vias and contact holes with high aspectratios. The presence of a robust etch endpoint determination signal asin any of the embodiments disclosed above thus allows for a shorter etchtime, and, hence, increased throughput, compared to conventional etchingprocesses.

Thus, embodiments of the present invention fill a need in present dayand future technology for optimizing selection of wavelengths to monitorfor endpoint determination or detection during etching. Similarly,embodiments of the present invention fill a need in present day andfuture technology for being able to determine an etch endpointexpeditiously, robustly, reliably and reproducibly under a variety ofdifferent conditions, even in real-time processing.

Further, it should be understood that the present invention isapplicable to any plasma etching system, including reactive ion etching(RIE), high-density, inductively coupled plasma (ICP) systems, electroncyclotron resonance (ECR) systems, radio frequency induction systems,and the like.

The particular embodiments disclosed above are illustrative only, as theinvention may be modified and practiced in different but equivalentmanners apparent to those skilled in the art having the benefit of theteachings herein. Furthermore, no limitations are intended to thedetails of construction or design herein shown, other than as describedin the claims below. It is therefore evident that the particularembodiments disclosed above may be altered or modified and all suchvariations are considered within the scope and spirit of the invention.Accordingly, the protection sought herein is as set forth in the claimsbelow.

What is claimed:
 1. A method for determining an etch endpoint, themethod comprising: collecting intensity data representative of opticalemission spectral wavelengths during a plasma etch process; analyzing atleast a portion of the collected intensity data into at most first andsecond Principal Components with respective Loadings and correspondingScores; and determining the etch endpoint using the respective Loadingsand corresponding Scores of the second Principal Component as anindicator for the etch endpoint using thresholding applied to therespective Loadings of the second Principal Component.
 2. The method ofclaim 1, wherein determining the etch endpoint includes using a trace ofScores corresponding to at least one of the first and the secondPrincipal Components along time to signal the etch endpoint.
 3. Themethod of claim 2, wherein using the trace of the Scores correspondingto the at least one of the first and the second Principal Componentsalong time to signal the etch endpoint includes using the trace of theScores corresponding to the second Principal Component along time tosignal the etch endpoint.
 4. The method of claim 1 wherein determiningthe etch endpoint further comprises: selecting multiple wavelengths asindicator variables based on the respective Loadings of at least one ofthe first and the second Principal Components, the multiple wavelengthsvarying during the plasma process so that the etch endpoint can bedetermined by monitoring the multiple wavelengths.
 5. The method ofclaim 2 wherein determining the etch endpoint further comprises:selecting multiple wavelengths as indicator variables based on therespective Loadings of the second Principal Component, the multiplewavelengths varying during the plasma process so that the etch endpointcan be determined by monitoring the multiple wavelengths.
 6. The methodof claim 3 wherein determining the etch endpoint further comprises:selecting multiple wavelengths as indicator variables based on therespective Loadings of the second Principal Component, the multiplewavelengths varying during the plasma process so that the etch endpointcan be determined by monitoring the multiple wavelengths.
 7. The methodof claim 1 further comprising mean-scaling the at least the portion ofthe collected intensity data prior to analyzing the at least the portionof the collected intensity data.
 8. The method of claim 1, whereinanalyzing the at least the portion of the collected intensity data intothe at most first and second Principal Components includes using aneigenanalysis method.
 9. The method of claim 1, wherein analyzing the atleast the portion of the collected intensity data into the at most firstand second Principal Components includes using a singular valuedecomposition method.
 10. The method of claim 1, wherein analyzing theat least the portion of the collected intensity data into the at mostfirst and second Principal Components includes using a power method. 11.A method for etching a wafer, the method comprising: etching a waferusing a plasma process so that a light-emitting discharge is produced;terminating the etching of the wafer when an etch endpoint isdetermined, wherein the determination of the etch endpoint furthercomprises: collecting intensity data representative of optical emissionspectral wavelengths during a plasma etch process; analyzing at least aportion of the collected intensity data into at most first and secondPrincipal Components with respective Loadings and corresponding Scores;and determining the etch endpoint using the respective Loadings andcorresponding Scores of the second Principal Component as an indicatorfor the etch endpoint using thresholding applied to the respectiveLoadings of the second Principal Component.
 12. The method of claim 11,wherein determining the etch endpoint includes using a trace of Scorescorresponding to at least one of the first and the second PrincipalComponents along time to signal the etch endpoint.
 13. The method ofclaim 12, wherein using the trace of the Scores corresponding to the atleast one of the first and the second Principal Components along time tosignal the etch endpoint includes using the trace of the Scorescorresponding to the second Principal Component along time to signal theetch endpoint.
 14. The method of claim 11 wherein determining the etchendpoint further comprises: selecting multiple wavelengths as indicatorvariables based on the respective Loadings of at least one of the firstand the second Principal Components, the multiple wavelengths varyingduring the plasma process so that the etch endpoint can be determined bymonitoring the multiple wavelengths.
 15. The method of claim 12 whereindetermining the etch endpoint further comprises: selecting multiplewavelengths as indicator variables based on the respective Loadings ofthe second Principal Component, the multiple wavelengths varying duringthe plasma process so that the etch endpoint can be determined bymonitoring the multiple wavelengths.
 16. The method of claim 13 whereindetermining the etch endpoint further comprises: selecting multiplewavelengths as indicator variables based on the respective Loadings ofthe second Principal Component, the multiple wavelengths varying duringthe plasma process so that the etch endpoint can be determined bymonitoring the multiple wavelengths.
 17. The method of claim 11 furthercomprising mean-scaling the at least the portion of the collectedintensity data prior to analyzing the at least the portion of thecollected intensity data.
 18. The method of claim 11, wherein analyzingthe at least the portion of the collected intensity data into the atmost first and second Principal Components includes using aneigenanalysis method.
 19. The method of claim 11, wherein analyzing theat least the portion of the collected intensity data into the at mostfirst and second Principal Components includes using a singular valuedecomposition method.
 20. The method of claim 11, wherein analyzing theat least the portion of the collected intensity data into the at mostfirst and second Principal Components includes using a power method. 21.A method for determining an etch endpoint, the method comprising:collecting intensity data representative of optical emission spectralwavelengths during a plasma etch process; analyzing the collectedintensity data into at most first and second Principal Components; anddetermining the etch endpoint using Loadings and Scores corresponding tothe second Principal Component as an indicator for the etch endpointusing thresholding applied to the Loadings corresponding to the secondPrincipal Component.
 22. The method of claim 21, wherein determining theetch endpoint using the Scores corresponding to the second PrincipalComponent as an indicator includes using a trace of Scores of at leastone of the first and the second Principal Components along time tosignal the etch endpoint.
 23. The method of claim 22, wherein using thetrace of the Scores of the at least one of the first and the secondPrincipal Components along time to signal the etch endpoint includesusing the trace of the Scores of the second Principal Component alongtime to signal the etch endpoint.
 24. The method of claim 21 whereindetermining the etch endpoint using the Scores corresponding to thesecond Principal Component as an indicator further comprises: selectingmultiple wavelengths as indicator variables based on Loadings of atleast one of the first and the second Principal Components, the multiplewavelengths varying during the plasma process so that the etch endpointcan be determined by monitoring the multiple wavelengths.
 25. The methodof claim 22 wherein determining the etch endpoint using the Scorescorresponding to the second Principal Component as an indicator furthercomprises: selecting multiple wavelengths as indicator variables basedon the Loadings corresponding to the second Principal Component, themultiple wavelengths varying during the plasma process so that the etchendpoint can be determined by monitoring the multiple wavelengths. 26.The method of claim 23 wherein determining the etch endpoint using theScores corresponding to the second Principal Component as the indicatorfurther comprises: selecting multiple wavelengths as indicator variablesbased on the Loadings corresponding to the second Principal Component,the multiple wavelengths varying during the plasma process so that theetch endpoint can be determined by monitoring the multiple wavelengths.27. The method of claim 21 further comprising mean-scaling the collectedintensity data prior to analyzing the collected intensity data.
 28. Themethod of claim 21, wherein analyzing the collected intensity data intothe at most first and second Principal Components includes using aneigenanalysis method.
 29. The method of claim 21, wherein analyzing thecollected intensity data into the at most first and second PrincipalComponents includes using a singular value decomposition method.
 30. Themethod of claim 21, wherein analyzing the collected intensity data intothe at most first and second Principal Components includes using a powermethod.
 31. A method for etching a wafer, the method comprising: etchinga wafer using a plasma process so that a light-emitting discharge isproduced; terminating the etching of the wafer when an etch endpoint isdetermined, wherein the determination of the etch endpoint furthercomprises: collecting intensity data representative of optical emissionspectral wavelengths during a plasma etch process; analyzing thecollected intensity data into at most first and second PrincipalComponents; and determining the etch endpoint using Loadings and Scorescorresponding to the second Principal Component as an indicator for theetch endpoint using thresholding applied to the Loadings correspondingto the second Principal Component.
 32. The method of claim 31, whereindetermining the etch endpoint using the Scores corresponding to thesecond Principal Component as an indicator includes using a trace ofScores of at least one of the first and the second Principal Componentsalong time to signal the etch endpoint.
 33. The method of claim 32,wherein using the trace of the Scores of the at least one of the firstand the second Principal Components along time to signal the etchendpoint includes using the trace of the Scores of the second PrincipalComponent along time to signal the etch endpoint.
 34. The method ofclaim 31 wherein determining the etch endpoint using the Scorescorresponding to the second Principal Component as an indicator furthercomprises: selecting multiple wavelengths as indicator variables basedon Loadings of at least one of the first and the second PrincipalComponents, the multiple wavelengths varying during the plasma processso that the etch endpoint can be determined by monitoring the multiplewavelengths.
 35. The method of claim 32 wherein determining the etchendpoint using the Scores corresponding to the second PrincipalComponent as an indicator further comprises: selecting multiplewavelengths as indicator variables based on the Loadings correspondingto the second Principal Component, the multiple wavelengths varyingduring the plasma process so that the etch endpoint can be determined bymonitoring the multiple wavelengths.
 36. The method of claim 33 whereindetermining the etch endpoint using the Scores corresponding to thesecond Principal Component as the indicator further comprises: selectingmultiple wavelengths as indicator variables based on the Loadingscorresponding to the second Principal Component, the multiplewavelengths varying during the plasma process so that the etch endpointcan be determined by monitoring the multiple wavelengths.
 37. The methodof claim 31 further comprising mean-scaling the collected intensity dataprior to analyzing the collected intensity data.
 38. The method of claim31, wherein analyzing the collected intensity data into the at mostfirst and second Principal Components includes using an eigenanalysismethod.
 39. The method of claim 31, wherein analyzing the collectedintensity data into the at most first and second Principal Componentsincludes using a singular value decomposition method.
 40. The method ofclaim 31, wherein analyzing the collected intensity data into the atmost first and second Principal Components includes using a powermethod.
 41. A computer-readable, program storage device encoded withinstructions that, when executed by a computer, perform a method, themethod comprising: collecting intensity data representative of opticalemission spectral wavelengths during a plasma etch process; analyzing atleast a portion of the collected intensity data into at most first andsecond Principal Components with respective Loadings and correspondingScores; and determining the etch endpoint using the respective Loadingsand corresponding Scores of the second Principal Component as anindicator for the etch endpoint using thresholding applied to therespective Loadings of the second Principal Component.
 42. A computerprogrammed to perform a method, the method comprising: collectingintensity data representative of optical emission spectral wavelengthsduring a plasma etch process; analyzing at least a portion of thecollected intensity data into at most first and second PrincipalComponents with respective Loadings and corresponding Scores; anddetermining the etch endpoint using the respective Loadings andcorresponding Scores of the second Principal Component as an indicatorfor the etch endpoint using thresholding applied to the respectiveLoadings of the second Principal Component.